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Asi Bunyajitradulya
Department of Mechanical Engineering
Faculty of Engineering
Chulalongkorn University
A Systematic Approach to
Overview, Conduct, and Design of an Experiment:Part I: Defining an Experiment / Objective / Experimental Condition / Scope
Part II: DRD (partial)
What is this lecture about?
A systematic approach to
overview, conduct, and design of an experiment
Contents
Goal of an experiment
Background and motivation to the systematic approach based on
ILL-defined problem VS Well-defined problem,
and the roles of different variables in a problem:
);;( cpxfy
);;( cpxfy
Typical engineering problems are related to the question of
whether and, if so, how does y vary with x under the
condition of various p and constant c?
);;( cpxfy
Contents
Practice: Identifying
from familiar engineering relations, graphs, tables
Recognizing the underlying condition/assumption [especially c
in y = f ( x ; p ; c ) ] of these relations, graphs, tables
Defining an experiment:
Three-Column Objective
Definition of an experiment/objective
Experimental condition
Scope of an experiment
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);;( cpxfy
This much is expected of you at the end of the hour.
Contents
Practice: Setting up the objective of an experiment (defining an experiment),
stating the experimental condition and the scope of an
experiment
Data Reduction Diagram (DRD):
The mechanic for the design and conduct of an experiment
Summary
This much is expected of you at the end of the hour.
This much is expected of you to know at the end of the hour.
But to do, may be some time later.
Goal of an Experiment
Goal of An Experiment for physical sciences / physical systems
Extract knowledge and useful information
regarding certain aspects of the physical system of interest
with reasonable justification and high level of confidence (that it is
reasonably true and accurate)
justification = approach/method + supporting evidences
Uses of the results of an experiment
That knowledge can be used for,
design and product development (data for the)
determine Young’s modulus of structural steel at standard condition
determine power-rpm relation for the newly designed engine
product testing and qualification according to some standard
test an air conditioner whether it qualifies for energy efficiency
qualification of an instrument
calibrate an instrument and quantify its performance parameters, e.g., accuracy, etc.
development of a mathematical model for a physical system (data for the)
empirical coefficients in many mathematical models
determine the scope and the level of accuracy of a theory (“verification/ falsification”)
“verify” beam deflection theory
etc.
Background and motivation for the systematic approach based on
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Typical engineering problems are related to the question of
whether and, if so, how does y vary with x under the
condition of various p and constant c?
Is there really a (truly-constant) constant?
Is gravitational acceleration g a constant,
g = 9.81 m/s2 ?
It may be a constant in this work of yours,
but how about your next work?
g depends on the condition: g = f (h,…)In this case, elevation h, and ….?
relation or function
Is there really a (truly-constant) constant?
Is Young’s modulus of structural steel a constant,
E = 200 GPa?
It may be a constant in this work of
yours,
but how about your next work?
NIST Report on the properties of structural steel from World Trade Center collapsesFigure from Luecke, et al., 2005, Federal Building and Fire Safety Investigation of The World Trade Center Disaster: Mechanical Properties of Structural Steels, NIST NCSTAR1-3D, http://fire.nist.gov/bfrlpubs/fire05/PDF/f05158.pdf
E depends on the condition: E = f (Material, T, …)
relation or function
Similar problem with “constant”
Is the “performance” of a car (the same car) the same as I drive in
Bangkok vs Chiang Mai ?
Bangkok vs New York ?
To put it more physically, i.e., in terms of physical quantities:
hot and dry day
vs hot and humid day
vs cold and dry day
vs cold and humid day?
“Per” depends on the condition
Similar problem with “constant”
To put this in even more workable engineering terms:
Whether and, if so, how does intake air temperature (T) affect the “performance =
power P” of an engine of a car …?
P = f (T,…) Whether and, if so, how does intake air humidity ratio () affect the “performance =
power” of an engine of a car …?
P = f (,T,…)
“Per” depends on the condition
Relation / Function of many variables: y = f (x1, x2, …)
Problem with “constant”
Problem with “constant”
It is rare that you will encounter a true/universal constant in engineering work.
The physical quantity of interest q depends on other physical quantities:
q1, q2, … through a physical relation:
Physical Quantities (PQ) and Physical Relations (PR)
Because of PR, this specific number is valid only under certain condition.
It may be constant in this work of yours, but how about next work of yours?
,...),( 21 qqfq
Typical Engineering Questions (not yet complete)
Whether and, if so, how does y vary with x ……?
)....;( xfy
y = dependent variable
x = independent variables
ILL-defined problem
VS Well-defined problem
The roles of different variables in a problem:
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ILL-Defined Problem The problem of the definition of the problem
Whether and how does the (specific) volume v of a gas (Helium)
vary with its temperature T?
The problem is not well-defined.
Depending on the condition, e.g.,
constant pressure, T v
if compressed fast enough, T v
....);(
)....;(
Tfv
xfy
freely moving lid compressiony = v
x = T
Helium
y = dependent variable
x = independent variables
Whether and how does specific volume v vary with temperature T
under the condition of constant pressure P = P1 for a gas Helium (R)?
The problem is now well-defined.
),;;(
);;(
RPTfv
cxfy
y = v
x = T
c = [R=Ro( Helium), P=P1]
P=P1
);;( cxfy
freely moving lid
P = P1
y = dependent variable
x = independent variables
…
c = constant parameters
Well-defined Problem and The Roles of Different Variables in a Problem:
Well-defined Problem and The Roles of Different Variables in a Problem:
Whether and how does specific volume v vary with temperature T
under the condition of various constant pressures P = P1 , P2, P3, for a gas
Helium (R)?
);;(
);;(
RPTfv
cpxfy
y = v
x = T
c = R (=Ro, Helium)
);;( cpxfy
P=P1
freely moving lid
P = P1
freely moving lid
P = P2
P=P2
freely moving lid
P = P3
P=P3
p = P
y = dependent variable
x = independent variables
p = variable parameters
c = constant parameters
Typical Engineering Questions
Whether and how does y vary with x under the condition
of various p and constant c?
);;( cpxfy
y = dependent variable
x = independent variables
p = variable parameters
c = constant parameters
Convention for the functional form: );;( cpxfy
y = dependent variable
x = independent variables
p = variable parameters
c = constant parameters
x
y c = co
p = p1
p = p2
p = p3
p
Graphical representation
c = co
p = p1 p = p2 p = p3
x y y y
. . . .
. . . .
. . . .
Tabular representation
);;( cpxfy
),...,;,...,;,...,( ccppxxfy
semicolons
More Examples
Problem 1:
Problem 2:
);;( RPTfv
);;( RTPfv
T
v R
P = P1
P = P2
P = P3 What process is this?
Isobaric
P
v R
T = T1
T = T2
T = T3 What process is this?
Isothermal
More Examples
Problem 3:
Problem 4:
);;( PRTfv
T
v P=Patm
R = R1 (air)
R = R2 (Helium)
R = R3 (Hydrogen) What process is this?
Isobaric
)substanceoftype;;,( waterTvfP
Figure from http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/pvtexp.html
Water
Practice
Identifying
from familiar engineering relations, graphs, tables
Recognizing the underlying condition/assumption [especially c in y
= f ( x ; p ; c ) ] of these relations, graphs, tables
);;( cpxfy
Example 1: Identify );;( cpxfy
Figure from Fox, R. W., Pritchard, P. J., and McDonald, A. T., 2008, Fluid Mechanics, Seventh Edition, Wiley, New York.
);;( atmPPfluidsoftypeTf
NOTE 1:
y, x, p, c can be type, state, condition, e.g.,
• Type of fluids
• Type of beam supports
• simply support, cantilever, etc.
• State of flows
• laminar, turbulent
Example 2: Identify );;( cpxfy
Figure from Fox, R. W., Pritchard, P. J., and McDonald, A. T., 2008, Fluid Mechanics, Seventh Edition, Wiley, New York.
Developing state of flow
)flowdevelopedfully
flowofstatedeveloping;flowtheofstate/,/;Re(
TLDeff
Laminar/Turbulent state of flow
Example 3: Identify );;( cpxfy
Table from http://depts.washington.edu/matseed/mse_resources/Webpage/Bicycle/Bicycle%20Materials%20Case%20Study.htm
);;( cpxfy
),,;;( STPrefSTPref TTPPSteelCarbonMediumMaterialfE
),;;( STPrefSTPref TTPPMaterialfE
NOTE 2:
x, p, c slots can be empty
[but usually not all empty at the same time.]
);;(
);;(
cfy
cxfy
How does
affect the design of an experiment,
especially the test rig?
);;( cpxfy
How does affect the design of an experiment, especially the test rig?
Problem 1:
Problem 2:
);;( kmxfa
x
|a| k=ko
m = m1
m = m2
m = m3
m x
Equilibriumk
Design of experiment and test rig:
change mass, fix spring
);;( mkxfa
Design of experiment and test rig:
change spring, fix mass
x
|a| m=mo
k = k1
k = k2
k = k3
);;( cpxfy
Defining an experiment:
Three-Column Objective
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Question and Definition of An Experiment
Question: Whether and how does y vary with x
under the condition of various p and constant c ?
Definition of an Experiment:
y, x, p, and c play different roles in our problem.
);;( cpxfy y = dependent variable
x = independent variables
p = variable parameters
c = constant parameters
Whether and how does y vary with x under the condition of various p and constant c ?
NOTE: The question of Whether (and correlation study)
If we do not know that x affects y (or x and y are related) or not, especially in complex systems
where there are many and random factors involved, for example:
Whether there is a relation between GPAX of first-year students (x) and GPAX of the students
when they graduate (y)?
Whether the pill x for curing disease z has the side effect y on patients of disease z?
This kind of study is usually referred to as ‘correlation study.’
In correlation study, typically we need to find the correlation coefficient between y and x in order to
answer the question whether y and x are related.
);;( cpxfy
Defining an experiment with the three-column objective
Objective Statement
(Question)
Objective Functional Form Objective Graphical
Representation
The effect of x on y under
the condition of various p
and constant c ….
);;( cpxfy y
x
c=co
p
p =p1
p=p2
p=p3
Defining an experiment with the three-column objective
Formulate clearly a well-defined problem.
[List all relevant variables and/or conditions: y, x, p, and c. Use functional form:
]
We know what to plot right from the beginning once we formulate our problem. NOT collect data first, then think
what to plot later.
We know and can outline how to extract results (or answer to your question posted in the objective) right from the
beginning. [From the graphical presentation of .]
Formulate hypotheses:
What does it mean when y increase/decrease with x?
What does it mean if it has or has no local minimum/maximum?
Objective Statement
(Question)
Objective Functional Form Objective Graphical
Representation
…. ….);;( cpxfy
);;( cpxfy
);;( cpxfy
Well-defined = Well-defined system
Well-defined
= Well-defined system and well-defined
problem/question regarding the system
All relevant variables: y, x, p, and c, must be accounted
for.
);;( cpxfy
);;( cpxfy
Convention: - Semicolonned slots);;( cpxfy
y = dependent variable
x = independent variables
p = variable parameters
c = constant parameters
x
y c = co
p = p1
p = p2
p = p3
p
Graphical representation
c = co
p = p1 p = p2 p = p3
x y y y
. . . .
. . . .
. . . .
Tabular representation
);;( cpxfy
),...,;,...,;,...,( ccppxxfy
semicolons
Definition of an experiment/objective
Experimental condition
Scope of an experiment
Defining an Experiment/ObjectiveDefinition, Experimental Condition, Scope, and Resolution
Definition of an experiment/objective:
Experimental Condition:
Scope of an experiment:
Range and resolution of each x and p, the value of each c
);;( cpxfy
);( cp
);with;with( 2121 occppppxxxx
Various types of questions/objectives in an experiment
Many variables in any
one slot
No variable parameter
Many questions
Fixed condition
Etc.
),...,;,...,;,...,( oo ccccppxxfy
);;( occxfy
);;(
);;(
);;(
cpxfy
cpxfy
cpxfy
);;( cfy
…….
Practice
Setting up the objective of an experiment (defining an experiment),
stating the experimental condition and the scope of an experiment
1. Identify
2. State the objective
3. State the experimental condition
4. Specify the scope
of the following experiments
);;( cpxfy
Practice Problem: Experiment 1
Figure from Fox, R. W., Pritchard, P. J., and McDonald, A. T., 2008, Fluid Mechanics, Seventh Edition, Wiley, New York.
.
Objective: To investigate the effect of temperature [x =T] on absolute
viscosity [y =] of various fluids [p = type of fluids] under the
condition of a fixed pressure at atmospheric [c = P (= Patm)].
Experimental Condition: for various fluids (…, …, …)
and at a fixed pressure (at atmospheric)
Scope: over temperature range of -20 <= T <= 120 oC, T = … oC
fluids tested are hydrogen, air,…..
at a fixed pressure at atmospheric
);;( PfluidsofTypeTf
Practice Problem: Experiment 2
Figure from Fox, R. W., Pritchard, P. J., and McDonald, A. T., 2008, Fluid Mechanics, Seventh Edition, Wiley, New York.
.
Objective: To investigate the effect of Reynolds number [x = Re] on the friction factor [y = f ]
in pipe flows for various relative roughness [p = e/D] and two states of the flow [p2 = L/T state
of the flow]: laminar and turbulent, under the condition of fully-developed flows [c = developing
state of flow].
Experimental Condition: various relative roughness (p = e/D)
for both laminar and turbulent flows (p2 = L/T state of the flow)
under the condition of fully-developed flows
Scope: over Reynolds number range of 500 <= Re <= 8x108, Re = …
over relative roughness range of 0 (smooth pipe) <= e/D <= 0.05, (e/D) = …
laminar and turbulent flows
fully-developed flow only
)flowdevelopedfully
flowofstatedeveloping;flowtheofstate/,/;Re(
TLDeff
Practice Problem: Experiment 3
Properties of water
Figure From Fox, R. W., Pritchard, P. J., and McDonald, A. T., 2008, Fluid Mechanics, Seventh Edition, Wiley, New York.
Practice Problem: Experiment 4 (red) & 5 (blue)
Table from http://depts.washington.edu/matseed/mse_resources/Webpage/Bicycle/Bicycle%20Materials%20Case%20Study.htm
Data Reduction Diagram (DRD)
The mechanic for the design and conduct of an experiment
Data Reduction Diagram (DRD)
);;( cpxfy
x
y c = co
pcp
yx,
),(
How exactly do you get the numerical values
for the coordinates in your
experiment?
cpyx
,),(
cp
xy
DRD,DRD
,DRD,DRD
DRD of the experiment
Construct DRD for each and every quantity in your objective functional form.
The underlying idea of DRD is
We must be able to trace each and every numerical transformation in our
experiment, exactly as we do in our experiment,
from the sources (the bottomost level boxes) to the final quantity at the top (the
top box).
);;( cpxfy
cDRDpDRDxDRDyDRD
Example: DRD’s of an experiment/objective
)flowdevelopedfully
flowofstatedeveloping;flowtheofstate/,/;Re(
);,;( 21
TLDeff
cppxfyObjective:
DRD - f
][)/(
2
1),,,,(
2essDimensionl
DLV
pDLVpf
Bottommost level boxes
DRD - Re
][),,(Re essDimensionlDV
DV
Bottommost level boxes
DRD.. DRD.. DRD..
DRD of the experiment/objective
refers to DRD’s of all variables in y, x, p, and c-slots in
the objective.
);;( cpxfy
);;( cpxfy
cDRDpDRDxDRDyDRD
Measured Quantities VS Derived Quantities
In physical science, there are two and only two types of
quantities
Measured Quantities
Derived Quantities
The reason is that we don’t want anybody to just make up
any number for a physical quantity at their will.
Measured Quantity VS Derived Quantity
Measured Quantity: A quantity whose numerical value in the current
experiment is obtained/read from an instrument in
the unit of that instrument directly (no unit conversion).
Derived Quantity: A quantity whose numerical value in the current
experiment is derived from
a (valid) physical relation:
[be it in the form of an equation, graph, table, etc., or unit
conversion relation], and
the values of other quantities in the relation .
),...,( 21 xxfy
Referenced Quantities
In our experiment, we may not be able to measure q.
We then refer its numerical value from a reliable source/reference.
Recognize that, even then, someone somewhere must either measure it or
derive it.
Data Reduction Diagram (DRD)
Bottommost Level
All measured (or referenced) quantity boxes
Upper Level
All derived quantity boxes
The idea of DRD is
• We must be able to trace each and every numerical transformation in our experiment, exactly as we do in our experiment,
• from the sources (the bottomost level boxes) to the final quantity at the top (the top box).
Examples of Derived Quantity Boxes
Examples of Measured Quantity Boxes
Down to instrument identity
Example of Referenced Quantity Boxes
Down to the page number.
The key idea is “how do you know (in your experiment)?”
DRD – L/T State of Flow
L/T State of Flow {Visual observation at the jet exit}
Use(fulness) of DRD:
In the Design Stage
Roadmap: Roadmap for our experiment
All Measured Quantities: Know all measured quantities from all bottomost
boxes of all DRD’s of the experiment
All Derived Quantities: Know all derived quantities from all derived
boxes of all DRD’s of the experiment
All Underlying Assumptions: Know all underlying assumptions of our experiment
Instruments: Know all necessary instruments to be used in our
experiment Choose/Select instruments
DCW: Construct Data Collection Worksheet (DCW) – All bottomost boxes
DAW: Construct Data Analysis Worksheet (DAW) – All derived boxes
);;( cpxfy
Use(fulness) of DRD:
In the Design Stage
Design-Stage Uncertainty Analysis: Roadmap for design-stage uncertainty
analysis
Selecting/Choosing Instruments : Selecting/Choosing all instruments
such that our final results have
uncertainties within the
desired/specified levels.
);;( cpxfy
Use(fulness) of DRD:
In the Qualification Stage
Diagnostic Roadmap
In the Conduct / Final Stage
Analysis Roadmap: Roadmap for analyzing data
Diagnostic Roadmap: If something does not look right, we can trace
things
1) right from the beginning/sources (of numerical values),
2) through all the analyses and assumptions, and
3) to the ends.
Final Uncertainty Analysis
);;( cpxfy
Summary
Summary
Experimental Condition:
Scope of an experiment:
Range and resolution of each x and p, the value of each c
Objective Statement
(Question)
Objective Functional Form
(Definition of an experiment)
Objective Graphical
Representation
The effect of x on y under the
condition of various p and
constant c ….
….
cDRDpDRDxDRDyDRD
);;( cpxfy
);( cp
);with;with( 2121 occppppxxxx
y
x
c=co
p
p =p1
p=p2
p=p3
Summary
cDRDpDRDxDRDyDRD
);;( cpxfy
Upper Levels
All derived quantity boxes
Bottommost Level
All measured quantity boxes
Summary
All things in an experiment go back to
Goal of an experiment (knowledge with high level of confidence), and
Objective of an experiment:
For example:
Experimental condition and scope:
What is the scope of validity of your answer to ?
DRD’s:
How exactly do you get numerical values for each y, x, p, and c in ?
Approach: experimental setup (test rig + instrument):
How do you get the answer to ?
Uncertainties
How accurate is your answer to ?
);;( cpxfy
);;( cpxfy
);;( cpxfy
);;( cpxfy
);;( cpxfy
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