ENGR 2194 | Laboratory Notebook
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Contents
Laboratory 1 – Application of Algebra in Engineering: The One-Loop Circuit .............................................. 1
1.1 Laboratory Objective........................................................................................................................... 1
1.2 Educational Objectives ........................................................................................................................ 1
1.3 Background ......................................................................................................................................... 2
1.3.1 Ohm’s Law .................................................................................................................................... 2
1.3.2 Kirchhoff’s Voltage Law ................................................................................................................ 2
1.3.3 Resistors Connected in Series ...................................................................................................... 3
1.3.4 Resistors Connected in Parallel .................................................................................................... 3
1.3.5 Equipment .................................................................................................................................... 4
1.4 Procedure ............................................................................................................................................ 8
1.4.1 Circuit Number 1 .......................................................................................................................... 8
1.4.2 Circuit Number 2 .......................................................................................................................... 9
1.4.3 Circuit Number 3 ........................................................................................................................ 10
1.4.4 Data Analysis .............................................................................................................................. 11
1.5 MATLAB Commands .......................................................................................................................... 12
1.6 General Guidelines (Executive Summary) ......................................................................................... 13
1.7 One-Loop Circuit Grading Guidelines (Executive Summary)............................................................. 14
Laboratory 2 – Trigonometric Relationships: One and Two-Link Planar Robots ........................................ 15
2.1 Laboratory Objective ......................................................................................................................... 15
2.2 Educational Objectives ...................................................................................................................... 15
2.3 Background ....................................................................................................................................... 16
2.3.1 Reference Angle ......................................................................................................................... 16
2.3.2 Law of Cosines ............................................................................................................................ 16
2.3.3 Law of Sines ................................................................................................................................ 17
2.4 Procedure .......................................................................................................................................... 18
2.4.1 One-Link Robot........................................................................................................................... 18
2.4.2 Two-Link Robot .......................................................................................................................... 20
2.4.3 Data Analysis .............................................................................................................................. 21
2.5 New MATLAB Commands ................................................................................................................. 24
2.6 General Guidelines (Executive Summary) ......................................................................................... 25
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2.7 One and Two-Link Planar Robots Grading Guidelines (Executive Summary) ................................... 26
Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals ...................... 27
3.1 Laboratory Objective ......................................................................................................................... 27
3.2 Educational Objectives ...................................................................................................................... 27
3.3 Background ....................................................................................................................................... 28
3.3.1 Sinusoid Characteristics ............................................................................................................. 28
3.3.2 Sinusoidal Relationships ............................................................................................................. 28
3.3.3 Equipment .................................................................................................................................. 29
3.4 Procedure .......................................................................................................................................... 31
3.4.1 Sinusoidal Measurements – Part 1 ............................................................................................ 31
3.4.2 Sinusoidal Measurements – Part 2 ............................................................................................ 35
3.4.3 Sinusoidal Measurements – Part 3 ............................................................................................ 40
3.4.4 Data Analysis .............................................................................................................................. 44
3.5 New MATLAB Commands ................................................................................................................. 46
3.6 General Guidelines (Executive Summary) ......................................................................................... 47
3.7 Sinusoids in Engineering Grading Guidelines (Executive Summary) ................................................. 48
Laboratory 4 – Systems of Equations in Engineering: The Two-Loop Circuit ............................................. 49
4.1 Laboratory Objective ......................................................................................................................... 49
4.2 Educational Objective ....................................................................................................................... 49
4.3 Background ....................................................................................................................................... 50
4.3.1 Problem Statement .................................................................................................................... 50
4.3.2 Matrix Inverse Method .............................................................................................................. 50
4.3.3 Cramer’s Rule ............................................................................................................................. 50
4.3.4 Substitution ................................................................................................................................ 50
4.3.5 MATLAB ...................................................................................................................................... 51
4.4 Procedure .......................................................................................................................................... 52
4.4.1 Two-Loop Circuit ........................................................................................................................ 52
4.4.2 Data Analysis .............................................................................................................................. 53
4.5 New MATLAB Commands ................................................................................................................. 54
4.6 General Guidelines (Executive Summary) ......................................................................................... 55
4.7 Systems of Equations Grading Guidelines (Executive Summary)...................................................... 56
Laboratory 5 – Derivatives in Engineering: Velocity and Acceleration in Free-Fall .................................... 57
iv
5.1 Laboratory Objective ......................................................................................................................... 57
5.2 Educational Objective ....................................................................................................................... 57
5.3 Background ....................................................................................................................................... 58
5.4 Procedure .......................................................................................................................................... 59
5.4.1 Sample Rate #1 (Default Settings).............................................................................................. 60
5.4.2 Sample Rate #2 (Open-Source Arduino Settings) ...................................................................... 62
5.4.3 Sample Rate #3 (MATLAB Arduino Settings) .............................................................................. 63
5.4.4 Data Analysis .............................................................................................................................. 63
5.5 General Guidelines (Executive Summary) ......................................................................................... 65
5.6 Derivatives in Engineering Grading Guidelines (Executive Summary) .............................................. 66
Laboratory 6 – Integrals in Engineering: Work and Stored Energy in a Spring ........................................... 67
6.1 Laboratory Objective ......................................................................................................................... 67
6.2 Educational Objective ....................................................................................................................... 67
6.3 Background ....................................................................................................................................... 68
6.4 Procedure .......................................................................................................................................... 69
6.4.1 Work and Stored Energy in a Spring .......................................................................................... 69
6.4.2 Data Analysis .............................................................................................................................. 69
6.5 General Guidelines (Executive Summary) ......................................................................................... 71
6.6 Integrals in Engineering Grading Guidelines (Executive Summary) .................................................. 72
Laboratory 7 – Differential Equations in Engineering: The Leaking Bucket ................................................ 73
7.1 Laboratory Objective ......................................................................................................................... 73
7.2 Educational Objectives ...................................................................................................................... 73
7.3 Background ....................................................................................................................................... 74
7.4 Procedure .......................................................................................................................................... 76
7.4.1 Leaking Bucket System ............................................................................................................... 76
7.4.2 Data Analysis .............................................................................................................................. 77
7.5 General Guidelines (Executive Summary) ......................................................................................... 79
7.6 Differential Equations in Engineering Grading Guidelines (Executive Summary) ............................. 80
Laboratory 8 – Differential Equations in Engineering: Spring-Mass Vibration ........................................... 81
8.1 Laboratory Objective ......................................................................................................................... 81
8.2 Educational Objective ....................................................................................................................... 81
8.3 Background ....................................................................................................................................... 82
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8.4 Procedure .......................................................................................................................................... 84
8.4.1 Displacement of Mass on One and Two Springs ........................................................................ 84
8.4.2 Oscillation - Maximum Weight (Six Masses and Cradle, 321 grams) and Two Springs ............. 84
8.4.3 Oscillation - Maximum Weight (Six Masses and Cradle, 321 grams) and One Spring ............... 85
8.4.4 Oscillation - Minimum Weight (Four Masses, 225 grams) and Two Springs ............................. 85
8.4.5 Oscillation - Minimum Weight (Four Masses, 225 grams) and One Spring ............................... 85
8.4.6 Data Analysis .............................................................................................................................. 85
8.5 General Guidelines (Executive Summary) ......................................................................................... 87
8.6 Integrals in Engineering Grading Guidelines (Executive Summary) .................................................. 88
vi
Laboratory 1 – Application of Algebra in Engineering: The OneLoop Circuit
1.1 Laboratory Objective
The objective of this laboratory is to illustrate linear and quadratic applications that are utilized in
engineering. Supplementary information includes basic MATLAB commands and functions.
1.2 Educational Objectives
After performing this experiment, students should be able to:
1. Perform basic algebraic manipulations with linear equations.
2. Perform basic algebraic manipulations with quadratic equations.
3. Measure and identify the relationship between voltage, current, and resistance.
4. Apply basic functions in MATLAB toward the solution of engineering equations using the
command window.
5. Use MATLAB for plotting data.
1
Laboratory 1 Application of Algebra in Engineering: The One-Loop Circuit
1.3 Background
It is essential all engineers have an understanding of the fundamental laws of electricity. Ohm’s Law and
Kirchhoff’s Voltage Law are two such rules that are presented in this lab. In addition, knowledge of the
equipment and instrumentation that employ these laws is comparably important. Some of these
include ammeters, voltmeters, watt-meters, breadboards, and circuitry components such as resistors.
The implementation of these instruments is introduced in this lab.
1.3.1 Ohm’s Law
+
+
VS
R
−
VR
−
I
Figure 1.1: An Electrical Circuit Consisting of a Voltage Source VS and Resistive Element R
Ohm’s Law is a linear equation stating the voltage across a resistor is equal to the current flowing
through that resistor multiplied by the value of that resistor. The following equation relates to Figure
1.1:
𝑉𝑅 = 𝐼 ∗ 𝑅
(1.1)
The value VR is the voltage across the resistor in volts (V), I is the current flowing through the resistor in
amperes (A), and R is the resistance in ohms (Ω).
1.3.2 Kirchhoff’s Voltage Law
Kirchhoff’s Voltage Law states that the sum of the voltage rise is equal to the sum of the voltage drop in
a circuit.
∑ Voltage Rise = ∑ Voltage Drop
Therefore, for the circuit shown in Figure 1.1:
𝑉𝑆 = 𝑉𝑅 = 𝐼 ∗ 𝑅
(1.2)
2
Laboratory 1 Application of Algebra in Engineering: The One-Loop Circuit
1.3.3 Resistors Connected in Series
To simplify a circuit, several resistors can be replaced by a single “equivalent” resistor (𝑅𝑒𝑞 ). In a series
circuit, current is the same through all the resistors. Therefore, the equivalent resistance is the sum of all
the resistors, or
𝑅𝑒𝑞 (𝑆𝑒𝑟𝑖𝑒𝑠) = 𝑅1 + 𝑅2 + 𝑅3
Equivalent Resistance (Series)
(1.3)
Current, I
V1
R1
Vbat
Current, I
V2
Vbat
Req (Series)
R2
V3
R3
Figure 1.2: Equivalent resistors in series
1.3.4 Resistors Connected in Parallel
In a parallel circuit, voltage is the same across every resistor. The total current is the sum of the currents
flowing through the individual resistors. For resistors in parallel, the equivalent resistance is the
following,
1
1
1 −1
𝑅𝑒𝑞 (𝑃𝑎𝑟𝑎𝑙𝑙𝑒𝑙) = (𝑅 + 𝑅 + 𝑅 )
Equivalent Resistance (Parallel)
1
2
(1.4)
3
I0
I0
Vbat
R1
R2
I1
Vbat
R3
I2
Req (Parallel)
I3
Figure 1.3. Equivalent resistors in parallel
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Laboratory 1 Application of Algebra in Engineering: The One-Loop Circuit
1.3.5 Equipment
A breadboard and resistors are two electrical components presented in this lab. The function of
breadboards along with identification of resistor values will be reviewed. In addition, three types of
measuring devices are introduced in this lab. These include an ammeter, voltmeter, and watt-meter.
Lastly, multiple power supplies will be utilized in the circuit construction.
1.3.5.1 Breadboard
A breadboard is a medium to prototype a circuit. Circuit components are attached to the breadboard by
inserting wires or leads into the small holes arranged in grids on the board. Since these components are
not soldered in place, the pieces can be removed and the circuit easily changed. A standard breadboard
is shown in Figure 1.4.
+ -
a b c d e
f g h i j
+ The vertical strips that
run the length of the
breadboard are
electrically connected.
These strips are
usually used for
power and ground
connections.
The horizontal holes
on each side of the
breadboard are
connected together.
Anything plugged into
these five holes will
be electrically
connected together
NOTE: The horizontal
holes on one side of
the breadboard don’t
connect to the other
side.
+ -
a b c d e
f g h i j
+ -
Figure 1.4: A standard breadboard layout
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Laboratory 1 Application of Algebra in Engineering: The One-Loop Circuit
Inside the breadboard are metal contacts that connect the holes. These metal contacts join clusters
of five holes together and are connected per the black arrows shown in Figure 1.4. These clusters
of five holes can be considered as one node.
1.3.5.2 Resistors
Resistors are electrical components that dissipate power by consuming current. This enables engineers
to regulate the amount of current allowed to flow into succeeding components in the circuit. All
resistors have a maximum power limit. The tiny resistors used in lab are quarter watt resistors. Due to
physical size limitations for printing, a standard for defining resistor values has been developed. For the
larger resistors, the value is printed right on the casing. This standard uses color coded bands, which in
conjunction with Table 1.1 below, yield the resistor value. Figure 1.5 shows an example of a typical
resistor defined by colored bands.
Figure 1.5: A 1000 Ω Resistor
The colored bands of this resistor correspond to Table 1.1 and Table 1.2. Reading from left to right, the
first two bands give the first two digits of the resistor value. The third colored band is the multiplier.
This value tells to what power of ten we multiply the first two digits. The resistor value in Figure 1.5 is
found using Tables 1.1 and 1.2 as follows:
•
•
•
•
The Brown band corresponds to a 1.
The Black band corresponds to a 0.
The Red band indicates multiplying by 102.
The next Red band indicates a tolerance of ±2%.
The value of this resistor is 10 times 102 resulting in 1000 Ω with a tolerance of ± 2%.
Table 1.1: Resistor Color Band Values
Number
0
1
2
3
4
5
6
7
8
9
Color
Black
Brown
Red
Orange
Yellow
Green
Blue
Violet
Grey
White
Table 1.2: Resistor Tolerance Color Band Values
Tolerance
Color
± 1%
Brown
± 2%
Red
± 5%
Gold
± 10%
Silver
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Laboratory 1 Application of Algebra in Engineering: The One-Loop Circuit
1.3.5.3 Ammeter
An ammeter is a device that measures the current flowing in a circuit. Since it measures a quantity
moving through the circuit, it must be connected in series as shown below in Figure 1.6.
Ammeter
+
Voltage
Source
−
R
Figure 1.6: Placement of an Ammeter in a Circuit
1.3.5.4 Voltmeter
A voltmeter is a device that measures the voltage potential across an electrical component. As a result,
it is placed in parallel with the component whose voltage drop is being measured. A voltmeter is used
to measure the voltage drop across the resistor in Figure 1.7.
+
Voltage
Source
−
R
V
Voltmeter
Figure 1.7: Placement of a Voltmeter in a Circuit
6
Laboratory 1 Application of Algebra in Engineering: The One-Loop Circuit
1.3.5.5 Watt-meter
A watt-meter is a device that measures the power used by an electrical component. The power
delivered or absorbed is given by some basic equations related by Ohm’s Law:
𝑃 = 𝑉𝐼 = 𝐼 2 𝑅 =
𝑉2
𝑅
(1.5)
A watt-meter functions by simultaneously measuring the current passing through and the voltage drop
across the component. In practice, this requires four connections to a circuit. The current nodes will be
connected in series while the voltage nodes will be connected in parallel. The watt-meter that is
connected in Figure 1.8 is set up to measure the power dissipated by the resistor R.
Wattmeter
+
+
Voltage
Source
−
−
−
V
+
R
Figure 1.8: Placement of a Wattmeter in a Circuit
7
Laboratory 1 Application of Algebra in Engineering: The One-Loop Circuit
1.4 Procedure
Follow the steps outlined below after the Lab Teaching Assistant has explained how to use the
laboratory equipment.
1.4.1 Circuit Number 1
Objectives:
 Measure and identify the relationship between voltage, current, and resistance.
o Recognize and operate equipment such as a breadboard, power supply and ammeter.
 Perform basic algebraic manipulations with linear equations.
o Use Ohm’s Law to solve for an unknown resistance.
Tasks:
1. Make sure the power supply and ammeter are both turned “off.”
2. The value of the resistor in Figure 1.9 is unknown. Construct the circuit with “unknown resistor
#1” and use the laboratory equipment to find this value.
R
+
VS
Ammeter
−
Figure 1.9: Circuit for Section 1.4.1
3. Have a member of the teaching team check your circuit before you turn on the power supply or
ammeter.
4. Measure and record the current values in Table 1.3. Wait to calculate the resistance.

NOTE: On the power supply, the current should be turned all the way right (clockwise) and
the voltage all the way left (counterclockwise). The ammeter should be set to the 20 mA
setting.
Table 1.3: Circuit 1 Measurements
Voltage VS
(volts)
0
5
Measured Current I
(A)
Calculated Resistance R
(Ω)
12
15
18
8
Laboratory 1 Application of Algebra in Engineering: The One-Loop Circuit
1.4.2 Circuit Number 2
Objectives:
 Measure and identify the relationship between voltage, current, and resistance.
o Recognize and operate equipment such as a breadboard, power supply and ammeter.
 Perform basic algebraic manipulations with linear equations.
o Use Ohm’s Law to solve for an unknown resistance.
o Recognize that voltage sources in series are additive.
Tasks:
1. Make sure the power supply and ammeter are both turned “off.”
2. The value of the resistor in Figure 1.10 is unknown. Construct the circuit with “unknown resistor
#2” and use the laboratory equipment to find this value.
R
+
VS
−
Ammeter
Additional Voltage Source V
+ V −
Figure 1.10: Circuit for Section 1.4.2
3. Have a member of the teaching team check your circuit before you turn on the power supply or
ammeter.
4. Measure and record the current values in Table 1.4. Wait to calculate the resistance.
 NOTE: On the power supply, the current should be turned all the way right, clockwise, and
the voltage all the way left, counterclockwise. The ammeter should be set to the 20 mA
setting.
Table 1.4: Circuit 2 Measurements
Voltage VS
(volts)
0
5
Add. Voltage V
(volts)
3
3
7
3
9
3
Measured Current I
(amps)
Calculated Resistance R
(Ω)
9
Laboratory 1 Application of Algebra in Engineering: The One-Loop Circuit
1.4.3 Circuit Number 3
Objectives:


Measure and identify the relationship between voltage, current, and resistance.
o Recognize and operate a breadboard, power supply, voltmeter and ammeter.
Perform basic algebraic manipulations with quadratic equations.
o Use the measured current to find the total power of the circuit.
Tasks:
1. Make sure the power supply, ammeter, and voltmeter are all turned “off.”
2. The value of the current flowing through the circuit in Figure 1.11 is unknown. Construct the
circuit using a one and two hundred ohm resistor. Use the laboratory equipment to find it.
100 Ω
200 Ω
+
+
VS
−
−
Ammeter
Voltmeter
Figure 1.11: Circuit for Section 1.4.3
3. Have a member of the teaching team check your circuit before you turn on the power supply,
ammeter, or voltmeter.
4. Measure and record the current/voltage values in Table 1.5. Wait to calculate the power.
 NOTE: On the power supply, the current should be turned all the way right, clockwise, and
the voltage all the way left, counterclockwise. The ammeter should be set to the 20 mA
setting.
Table 1.5: Circuit Three Measurements
Voltage VS
(volts)
Measured
Current I
(amps)
Measured
V200 Ω
(volts)
Calculated
Power P100 Ω
(watts)
Calculated
Power P200 Ω
(watts)
Calculated
Power Ptotal
(watts)
5
10
15
10
Laboratory 1 Application of Algebra in Engineering: The One-Loop Circuit
1.4.4 Data Analysis
For Circuit Number 1:
1. Calculate the resistance R in the last column of Table 1.3 using Ohm’s Law. (Pay close attention to
UNITS!)
𝑅=
𝑉𝑆
𝐼
2. Attach these hand calculations at the end of the lab.
3. Plot I vs. VS using MATLAB.
 NOTE: Standard notation is y vs. x. Place I on the y-axis and VS on the x-axis.
 NOTE: Use the MATLAB syntax that is given in Section 1.5.
4. Using MATLAB’s Basic Fitting tool, find the slope of the graph.
5. Attach the graph at the end of the lab.
For Circuit Number 2:
1. Calculate the resistance R in the last column of Table 1.4 using Ohm’s Law. (Pay close attention to
UNITS!)
𝑅=
𝑉𝑆 + 𝑉𝑅
𝐼
2. Plot I vs. VS in MATLAB using the syntax in Section 1.5.
3. Using MATLAB’s Basic Fitting tool, find the slope and x-intercept of the graph.
4. Attach the graph at the end of the lab.
For Circuit Number 3:
1. Use equation 1.5 to calculate the power across each resistor in Table 1.5. Then, use the following
quadratic equation to calculate the total power and record that value in the last column of Table
1.5. (Pay close attention to UNITS!)
𝑃𝑡𝑜𝑡𝑎𝑙 = 𝑃100 Ω + 𝑃200 Ω = 𝐼 2 ∗ 𝑅100 Ω + 𝑉200 Ω ∗ 𝐼

NOTE: The value for R in this equation is 100 Ω, not 200 Ω. The value for V in this equation
is the voltage drop across the 200 Ω resistor.
11
Laboratory 1 Application of Algebra in Engineering: The One-Loop Circuit
1.5 MATLAB Commands
x = [ ];

This command defines a row vector x. Place real numbers within the square brackets separated
by spaces or commas.
plot (x, y, ‘o’)

This command plots the data in vectors x and y and does not connect lines between each point.
xlabel (‘an appropriate label for the x-axis, with units in parentheses’)

This command adds a label to the x-axis of the plot. Type the label inside of the single
parentheses.
ylabel (‘an appropriate label for the y-axis, with units in parentheses’)

This command adds a label to the y-axis of the plot. Type the label inside of the single
parentheses.
title (‘an appropriate, descriptive title for the plot’)

This command adds a title to the plot. Type the label inside of the single parentheses.
NOTE: To fit a linear curve to the data and place the equation of that line on the plot:
1. On the figure, go to the “Tools” drop down menu.
2. Highlight “Basic Fitting”.
3. Check the “Linear” box.
4. Check the “Show Equations” box.
NOTE: The “help” command in MATLAB can be used to find a description and example for functions
such as plot, xlabel, ylabel, and title.

For example, type “help plot” in the command window to learn more about the plot
function.
12
Laboratory 1 Application of Algebra in Engineering: The One-Loop Circuit
1.6 General Guidelines (Executive Summary)
1. Write an executive summary for this lab using the Grading Guidelines on the following page.
2. Answer the following questions within the body of the executive summary.
a) To what component of circuit number one does the slope of plot one correspond?
i. NOTE: Remember y=m*x + b and I=V/R.
b) To what component of circuit number two does the x-intercept of plot two correspond?
i. NOTE: You will have to solve for the x-intercept using y=m*x + b.
c) Refer to circuits number one and two for the following questions:
i.
For circuit number one and two, the calculated R should be relatively close to what
value?
ii. By how much do these R values differ from the theoretical resistance as a percentage?
Clearly state which values were used for theoretical and experimental resistance.

NOTE: Calculate the percent error of circuits #1 and #2. Use the voltage case
which most differs from the R values you listed for the previous question.
% 𝐸𝑟𝑟𝑜𝑟 =
|𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 − 𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙|
𝑥 100
𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
iii. Is this within the tolerance of the resistor?

NOTE: Search online for a picture of the resistors you previously listed and
determine their tolerance.
d) How do the values for power across the two resistors compare? What are some reasons for
this?
3. Place Executive Summary, Appendix and Grading Guidelines in one word document and submit to
Carmen Dropbox. (10 Points will be deducted if the Grading Guidelines sheet is missing)
13
Laboratory 1 Application of Algebra in Engineering: The One-Loop Circuit
1.7 One-Loop Circuit Grading Guidelines (Executive Summary)
Section
Point
Breakdown
Header Information (See Tech. Comm. Guide)
Executive Summary
5
60
Clearly address the purpose of examining circuits and discuss the principles
that were addressed throughout the lab.
5
Describe the main aspects of the lab procedure with proper transitions
between each step of the lab.
10
Examine the results obtained in the team's study of circuits for each of the
tasks and explain the significance of the results. What did each of the tasks
teach the team about circuits?
20
Incorporate answers to the lab questions within the body of the executive
summary. (See Section 1.6.2)
20
Clearly discuss the obstacles faced throughout the lab and address how
they were overcome. Did the team have problems with wiring, did the
multi-meter malfunction? Etc.
3
Provide concrete examples of how to improve the lab procedure in order to
improve understanding of circuits.
2
Writing Style
10
Grammar: See Technical Communication Guide for recommendations.
5
Organization and Progression: make sure the summary has natural breaks,
ideas are properly separated by paragraphs and topic sentences, and ideas
are fluently connected.
5
Appendix
25
Tables 1.3, 1.4, and 1.5
6
Show hand calculations for Tables 1.3, 1.4, 1.5 (one example per equation)
6
MATLAB m-files for Table 1.3 and 1.4
4
MATLAB plots for Table 1.3 and 1.4
6
Lab Participation Agreement
3
Group Grade
Points
Earned
100
14
Laboratory 2 – Trigonometric Relationships: One and Two-Link
Planar Robots
2.1 Laboratory Objective
The objective of this laboratory is to learn basic trigonometric functions, conversion from rectangular to
polar form, and vice-versa.
2.2 Educational Objectives
After performing this experiment, students should be able to:
1. Identify the basic trigonometric functions.
2. Explain the concept of a unit circle and four quadrants.
3. Describe the concept of a reference angle.
4. Perform the polar to rectangular and rectangular to polar coordinate conversion.
5. Prove a few of the basic trigonometric identities.
15
Laboratory 2 Trigonometric Relationships: One ad Two-Link Planar Robots
2.3 Background
Trigonometry is a tool that mathematically forms geometrical relationships. The understanding and
application of these relationships are vital for all engineering disciplines. Relevant applications include
automotive, aerospace, robotics, and building design. This lab will outline a few common, but useful,
trigonometric relationships.
2.3.1 Reference Angle
A reference angle is an acute angle (less than 90°) that may be used to compute the trigonometric
functions of the corresponding obtuse angle (greater than 90°). Figure 2.1 shows the reference angle ϕ
(Greek letter phi) with respect to the angle ϑ (Greek letter theta).
y
y
y
ϑ
ϑ
ϕ
x
x
ϕ
ϑ
ϕ
x
(a) 𝜙 = 180° − 𝜃
(b) 𝜙 = 𝜃 − 180°
(c) 𝜙 = 180° − 𝜃
Figure 2.1: Reference Angle Calculations in Different Quadrants
The reference angle ϕ is calculated using the formulas shown in the captions of each corresponding
subfigure (i.e., labeled as “a” through “c”) of Figure 2.1. See Figure 3.2 (p. 62) of “Section 3.2” in the
Rattan/Klingbeil textbook for details.
2.3.2 Law of Cosines
The law of cosines is a method that helps to solve triangles. The three formulas of Equation 2.1 relate
the sides and interior angles of Figure 2.2.
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos(𝐴)
𝑏 2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 cos(𝐵)
(2.1)
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos(𝐶)
16
Laboratory 2 Trigonometric Relationships: One ad Two-Link Planar Robots
b
C
A
a
c
B
Figure 2.2: Law of Cosines Triangle
2.3.3 Law of Sines
The Law of Sines is another method that helps to solve triangles. Using the triangle of Figure 2.2,
Equation 2.2 relates the sides to the interior angles.
𝑎
𝑠𝑖𝑛 (𝐴)
=
𝑏
𝑠𝑖𝑛 (𝐵)
=
𝑐
𝑠𝑖𝑛 (𝐶)
(2.2)
17
Laboratory 2 Trigonometric Relationships: One ad Two-Link Planar Robots
2.4 Procedure
Follow the steps outlined below after the Lab Teaching Assistant has explained how to use the
laboratory equipment.
2.4.1 One-Link Robot
Objectives:
 Explain the concept of a unit circle and four quadrants.
o Recognize the unit circles importance when calculating reference angles.
 Describe the concept of a reference angle.
o Use reference angles to calculate X and Y.
 Perform the polar to rectangular and rectangular to polar coordinate conversion.
Tasks:
1. Using the boards in the lab, fill in Table 2.1. Pay close attention to the sign of your answer for all
values. Wait to do the calculations.
 NOTE: Refer to Figure 2.3 to measure X and Y.
 NOTE: Refer to Figure 2.1 to calculate the reference angle.
𝜋
 NOTE: To convert a value in degrees to radians, the multiplying factor is 180.
y -axis
r
Y
θ
x-axis
X
Figure 2.3: One Link Robot
18
Laboratory 2 Trigonometric Relationships: One ad Two-Link Planar Robots
Table 2.1: Polar to Rectangular Conversion
Angle θ (◦)
Measured
X (mm)
Measured
Y (mm)
Vector
Form
Xî + Yĵ
r
(mm)
30
100
45
100
90
100
135
100
180
100
225
100
270
100
Reference
Angle
(◦)
Reference
Angle
(radians)
Calculated
X (mm)
Calculated
Y (mm)
2. Using the boards in the lab, fill in Table 2.2. Wait to do the calculations.
Table 2.2: Rectangular to Polar Conversion
(X,Y)
Measured
ϑ ( °)
Reference
Angle (°)
Reference
Angle
(radians)
Calculated
ϑ (° )
Calculated
r (mm)
Polar
Form
(r, ϑ)
(85,50)
(70,70)
(0,100)
(−70,70)
(−100,0)
(-70,-70)
19
Laboratory 2 Trigonometric Relationships: One ad Two-Link Planar Robots
2.4.2 Two-Link Robot
Objectives:
 Perform the polar to rectangular and rectangular to polar coordinate conversion.
o Recognize the difference in calculations between one and two link robots.
Tasks:
1. Using the boards in the lab, fill in the Measured Values of Table 2.3 using Figure 2.4 as a guideline.
Wait to do the calculations.
y-axis
P(X,Y)
r2
r
Y
180 − θ2
P(x1,y1)
r1
α
β
θ2
θ1
x-axis
X
Figure 2.4: General Two-Link Robot
Table 2.3: Two Link Robot Components
θ1(◦)
0
0
30
30
180
270
360
θ2(◦)
0
90
45
60
0
30
90
X
Y
Measured Values
x1 y1 x2 = X - x1 y2 = Y - y1
Calculated Values
X
Y
20
Laboratory 2 Trigonometric Relationships: One ad Two-Link Planar Robots
2.4.3 Data Analysis
Objectives:
 Prove a few of the basic trigonometric identities.
o Identify the relationships between sine, cosine, tangent, and secant
o Use the Law of Cosines, Law of Sines and Pythagorean Theorem
Tasks:
One-Link Robot
1. Use Equations 2.3 to find the calculated X and Y values. Place the values in Table 2.1.
𝑋 = 𝑟 cos(𝜃)
(2.3)
𝑌 = 𝑟 sin(𝜃)
2. Use Equations 2.4 to find the calculated θ and l. Place the values in Table 2.2.
𝜃 = tan−1 (𝑌/𝑋)
(2.4)
𝑟 = √𝑋 2 + 𝑌 2
Identity Verification
An identity is a trigonometric relationship that is true for all permissible values of the variable(s). Many
times, trigonometric identities are used to simplify more complex problems.
1. Using MATLAB, fill in Tables 2.4 and 2.5.
 The first column of Table 2.4 comes from Table 2.2.
 Define this column as a vector in MATLAB and perform element by element calculations
on it to get the other columns.
 NOTE: All calculations should be done with MATLAB. No calculator use!
Table 2.4: Identity Verification Part 1
Calculated ϑ from
sin(ϑ)
Table 2.2 (°)
cos(ϑ) tan(ϑ) sec(ϑ)
21
Laboratory 2 Trigonometric Relationships: One ad Two-Link Planar Robots
Table 2.5: Identity Verification Part 2
𝑠𝑖𝑛(𝜃)
𝑐𝑜𝑠(𝜃)
𝑠𝑖𝑛2 (𝜃) + 𝑐𝑜𝑠 2 (𝜃)
1 + 𝑡𝑎𝑛2 (𝜃)
𝑠𝑒𝑐 2 (𝜃)
Two-Link Robot
1. Write a MATLAB code to calculate X and Y by adding the components of each link. Place the
values in Table 2.3. When calculating X and Y, recall the following equations from class:
𝑥1 = 𝑟1 cos(𝜃1 )
𝑦1 = 𝑟1 sin(𝜃1 )
𝑥2 = 𝑟2 cos(𝜃1 + 𝜃2 )
𝑦2 = 𝑟2 sin(𝜃1 + 𝜃2 )
𝑋 = 𝑥1 + 𝑥2
𝑌 = 𝑦1 + 𝑦2
Solve a Triangle Using Law of Cosines and Law of Sines
In some cases, the Law of Sines and Cosines must both be used to solve a triangle. Figure 2.4 is one such
case where the lengths l1 and l2 along with the final ending point P of the two links are known and the θ
values are not. Both laws are needed to solve this triangle.
1. Using the Pythagorean Theorem, the radius r is found by:
𝑟 = √𝑋 2 + 𝑌 2
(2.5)
2. Using the Law of Cosines, θ2 is found by the following equation:
𝑟 2 = 𝑟12 + 𝑟22 − 2𝑟1 𝑟2 cos(180 − 𝜃2 )
22
Laboratory 2 Trigonometric Relationships: One ad Two-Link Planar Robots
𝑟 2 −𝑟12 −𝑟22
)
−2𝑟1 𝑟2
𝜃2 = 180 − cos −1 (
(2.6)
3. Using the Law of Sines, α is found by the following equation:
𝑟
𝑟2
=
sin(𝜃2) sin(𝛼)
𝑟2 sin(𝜃2 )
)
𝑟
𝛼 = sin−1 (
(2.7)
4. θ1 is now found by the equation:
𝑌
𝛽 = tan−1 (𝑋)
(2.8)
𝜃1 = 𝛽 − 𝛼
(2.9)
5. Using Equations 2.5, 2.6, 2.7, 2.8, and 2.9, write a MATLAB code to fill in Table 2.6.
 NOTE: r1 = r2 = 50 mm.
 NOTE: Define X and Y as vectors containing all points below.
Table 2.6: Application of Sine and Cosine Laws
P(X,Y)
(55,75)
(75,60)
(15,63)
(32,14)
(71,70)
θ2(°)
α(°)
β(°)
θ1(°)
23
Laboratory 2 Trigonometric Relationships: One ad Two-Link Planar Robots
2.5 New MATLAB Commands
disp (‘text’)

This command displays text if the letters/numbers in parenthesis are within two single
quotation marks. For example, it can be used to display the header “Table 2.3.”
sind (angle in degrees)

This command calculates the sine of an angle in degrees.
cosd (angle in degrees)

This command calculates the cosine of an angle in degrees.
tand (angle in degrees)

This command calculates the tangent of an angle in degrees.
secd (angle in degrees)

This command calculates the secant of an angle in degrees.
NOTE: You will be required to perform mathematical operations between two row vectors. As shown
below, you must use a dot operator for multiplication, division, and exponentiation. For two row
vectors x and y, you will need to use the following notation:





Addition: x + y or y + x
Subtraction: x - y or y - x
Multiplication: x .* y or y .* x
Division: x ./ y or y ./ x
Exponent: x .^ y or y .^ x
NOTE: The “help” command in MATLAB can be used to find a description and example for functions
such as disp, sind, and cosd.

For example, type “help disp” in the command window to learn more about the display
function.
NOTE: Refer to section “1.5 MATLAB Commands” for previously discussed material that you may also
need in order to complete this lab.
24
Laboratory 2 Trigonometric Relationships: One ad Two-Link Planar Robots
2.6 General Guidelines (Executive Summary)
1. Write an executive summary for this lab using the Grading Guidelines on the following page.
2. Answer the following questions within the body of the executive summary:
a) Based on your results for Tables 2.4 and 2.5, write down the three trigonometric identities
that were verified.
b) State two examples of real-world objects that are similar to a two link robot and rotate
about two different points. Describe one of your example’s function and application in no
more than two to three sentences.
3. Place Executive Summary, Appendix and Grading Guidelines in one word document and submit to
Carmen Dropbox. (10 Points will be deducted if the Grading Guidelines sheet is missing)
25
Laboratory 2 Trigonometric Relationships: One ad Two-Link Planar Robots
2.7 One and Two-Link Planar Robots Grading Guidelines (Executive Summary)
Section
Point
Breakdown
Header Information (See Tech. Comm. Guide)
Executive Summary
5
40
Clearly address the purpose of examining one and two-link planar robots.
Discuss the principles that were addressed throughout the lab.
3
Briefly describe the main aspects of the lab procedure with proper
transitions between each step of the lab. If necessary, reference figures
from the procedure and attach them to the Appendix.
8
Overall, what did each of the tasks teach the team about basic
trigonometric functions, reference angles, or conversion from polar to
rectangular coordinates?
12
Incorporate answers to the lab questions within the body of the executive
summary. (See Section 2.6.2)
12
Clearly discuss the obstacles faced throughout the lab and address how
they were overcome. For instance, did the team have problems taking
measurements with the board?
3
Provide concrete examples of how to improve the lab procedure in order to
improve understanding of basic trigonometric functions.
2
Writing Style
10
Grammar: See Technical Communication Guide for recommendations.
5
Organization and Progression: make sure the summary has natural breaks,
ideas are properly separated by paragraphs and topic sentences, and ideas
are fluently connected.
5
Appendix
45
Tables 2.1, 2.2, 2.3, 2.4, 2.5, and 2.6
18
Hand Calculations for Tables 2.1 and 2.2 (one example per equation)
8
MATLAB m-files for Tables 2.3, 2.4, 2.5 and 2.6
8
MATLAB Output for Tables 2.3, 2.4, 2.5 and 2.6
8
Lab Participation Agreement
3
Group Grade
Points
Earned
100
26
Laboratory 3 – Sinusoids in Engineering: Measurement and
Analysis of Harmonic Signals
3.1 Laboratory Objective
The objective of this laboratory is to understand the basic properties of sinusoids and sinusoid
measurements.
3.2 Educational Objectives
After performing this experiment, students should be able to:
1. Discuss the properties of sinusoids.
2. Explain sinusoidal addition.
3. Obtain measurements using an oscilloscope.
4. Use a function generator.
27
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
3.3 Background
Sinusoids are sine or cosine waveforms that can describe many engineering phenomena. Any oscillatory
motion can be described using sinusoids. Many types of electrical signals such as square, triangle, and
sawtooth waves are modeled using sinusoids. Their manipulation incurs the understanding of certain
quantities that describe sinusoidal behavior. These quantities are described below.
3.3.1 Sinusoid Characteristics
Amplitude The amplitude A of a sine wave describes the height of the hills and valleys of a sinusoid. It
carries the physical units of what the sinusoid is describing (volts, amps, meters, etc.).
Frequency There are two types of frequencies that can describe a sinusoid. The normal frequency f is
how many times the sinusoid repeats per unit time. It has units of cycles per second (s-1) or Hertz
(Hz). The angular frequency ω is how many radians pass per second. Consequently, ω has units of
radians per second.
Period The period T is how long a sinusoid takes to repeat one complete cycle. The period is measured
in seconds.
Phase The phase φ of a sinusoid causes a horizontal shift along the t-axis. The phase has units of
radians.
Time Shift The time shift ts of a sinusoid is a horizontal shift along the t-axis and is a time measurement
of the phase. The time shift has units of seconds.
𝜋
NOTE: A sine wave and a cosine wave only differ by a phase shift of 90° or 2 radians. In reality, they are
the same waveform but with a different φ value.
3.3.2 Sinusoidal Relationships
𝑡𝑆
Figure 3.1: Sinusoid
28
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
The general equation of a sinusoid is given below and refers to Figure 3.1.
𝑥(𝑡) = 𝐴𝑠𝑖𝑛(𝜔𝑡 + 𝜙)
(3.1)
The angular frequency is related to the normal frequency by Equation 3.2.
𝜔 = 2𝜋𝑓
(3.2)
The angular frequency is also related to the period by Equation 3.3.
𝜔=
2𝜋
𝑇
(3.3)
By inspection, the normal frequency is related to the period by Equation 3.4.
𝑓 =
1
𝑇
(3.4)
The time shift is related to the phase (radians) and the frequency by Equation 3.5.
𝑡𝑆 =
∅
2𝜋𝑓
(3.5)
3.3.3 Equipment
3.3.3.1 Inductors
Inductors are electrical components that resist a change in the flow of current passing through them.
They are essentially coils of wire. Inductors are electromagnets too. They are represented in schematics
using the following symbol and physically using the following equipment (with or without exposed wire):
Figure 3.2: Symbol and Physical Example for Inductors
3.3.3.2 Capacitors
Capacitors are electrical components that store energy. This enables engineers to store electrical
energy from an input source such as a battery. Some capacitors are polarized and therefore have a
negative and positive plate. One plate is straight, representing the positive terminal on the device, and
29
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
the other is curved, representing the negative one. Polarized capacitors are represented in schematics
using the following symbol and physically using the following equipment:
Figure 3.3: Symbol and Physical Example for Capacitors
3.3.3.3 Data Acquisition (DAQ) System
The National Instruments (NI) Data Acquisition (DAQ) System for this lab provides analog input, analog
output, digital input and output, audio, power supplies and digital multi-meter functions in a compact
USB device. A prototyping board can be attached in order to build an electrical circuit. A USB cable is
used to connect the system to a computer. ELVISmx is the driver software installed on a computer that
supports and controls the DAQ.
3.3.3.4 Function Generator
A function generator is used to create different types of electrical waveforms over a wide range of
frequencies. The function generator used in the lab is part of the ELVISmx software. It generates
standard sine, square, and triangle waveforms. It also uses the analog output channel on the DAQ.
3.3.3.5 Oscilloscope
An oscilloscope is a type of electronic test instrument that allows observation of constantly varying
voltages, usually as a two-dimensional plot of one or more signals as a function of time. The
oscilloscope used in this lab is part of the ELVISmx software. It displays voltage data over time for the
analysis of one or two voltage measurements taken from the analog input channels of the DAQ. The
observed waveform can be analyzed for amplitude, frequency, time interval and more.
30
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
3.4 Procedure
Follow the steps outlined below after the Lab Teaching Assistant has explained how to use the
laboratory equipment.
3.4.1 Sinusoidal Measurements – Part 1
Objectives:
 Obtain measurements using an oscilloscope.
 Use a function generator.
o Apply different settings to obtain different sinusoids.
 Discuss the properties of sinusoids.
o Measure amplitude, period, and frequency.
Tasks:
1.
Using a 560 Ω resistor, create the circuit below in Figure 3.4.
a) For this circuit, AI 0+ is oscilloscope channel 1+ (green wire) and AO 0 is function generator
channel + (orange wire). All black wire is connected to ground, a common reference point
with zero voltage.
b) NOTE: The resistor and wires do not have to be in the exact configuration that is shown on
the breadboard in order for it to work.
31
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
- +
j i h g f
e d c b a
- +
- +
j i h g f
e d c b a
- +
Figure 3.4: Example Breadboard for Part 1
2.
Use the table below to understand what the connections mean:
Table 3.1: DAQ Terminal Signal Descriptions
32
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
3.
Connect the data acquisition (DAQ) board to the computer using the provided USB cable.
4.
Click the Start button in the bottom left corner of your computer screen. Search programs and
files for “elvis.” The exact program title is “NI ELVISmx Instrument Launcher.”
5.
Open the function generator (labeled as “FGEN”) and oscilloscope (labeled as “Scope”).
6.
Apply the following settings to the function generator and click “Run” to complete the first row
of measurements/calculations on the following page in Table 3.2.
Figure 3.5: Settings on Function Generator for Part 1 (a)
7.
Use the oscilloscope to find the peak-to-peak voltage (Vp-p) or two times the amplitude,
frequency (f), and period (T) of the sinusoidal function for the first row of Table 3.2.
a) Set “Timebase” to 500 us (where “us” means micro or 10-6 seconds).
b) Click “Run” and “Autoscale.”
c) Once a sinusoid appears, click “Stop.”
d) To find the period (T), click the box next to “Cursors On” in the bottom left corner of the
oscilloscope window. Using Figure 3.1 as a guide, drag cursor 1 and 2 to positions along the
sinusoid where corresponding x-intercepts exist.
8.
Close the oscilloscope and reopen it.
9.
Apply the following settings to the function generator and click “Run” to complete the second
row of measurements/calculations in Table 3.2.
33
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
Figure 3.6: Settings on Function Generator for Part 1 (b)
10. Repeat Step 7 from above but set “Timebase” to 100 us (where “us” means micro or 10-6
seconds).
11. Add your measured values to Table 3.2. Wait to do the calculations.
Table 3.2: Sinusoid Measurements for Part 1
Function Generator
Voltage Frequency
(Vp−p )
(Hz)
2.5
1000
3
5000
Oscilloscope (Measured)
2*A
f
T
(Vp−p )
(Hz)
(sec)
Calculated
ωf
ωT
(rad/sec) (rad/sec)
T
(sec)
34
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
3.4.2 Sinusoidal Measurements – Part 2
Objectives:
 Obtain measurements using an oscilloscope.
o Use inductors and capacitors to generate different sinusoids.
 Use a function generator.
o Apply different settings to obtain various sinusoids.
 Discuss the properties of sinusoids.
o Measure the amplitude, period, and frequency.
Tasks:
1. Use two 560 Ω resistors, a 47 mH inductor, and a 0.1 μF capacitor to create the circuit below in
Figure 3.7.
a) For this circuit, AI 0+ is oscilloscope channel 1+ (green wire) and AO 0 is function generator
channel + (orange wire). All black wire is connected to ground, a common reference point
with zero voltage. All blue wire is used to bridge/connect points from each half of the
circuit.
b) NOTE: The resistor and wires do not have to be in the exact configuration that is shown on
the breadboard in order for it to work.
35
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
- +
j i h g f
e d c b a
- +
- +
j i h g f
e d c b a
- +
Figure 3.7: Example Breadboard for Part 2 (a)
36
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
2.
Apply the following settings to the function generator and click “Run.”
Figure 3.8: Settings on Function Generator for Part 2
3.
Use the oscilloscope to find the peak-to-peak voltage (Vp-p), frequency (f), and period (T) of the
sinusoidal function for the signal across the capacitor (Vc). Complete the corresponding row of
Table 3.3.
a) Set “Timebase” to 100 us (where “us” means micro or 10-6 seconds).
b) Click “Run” and “Autoscale.”
c) Once a sinusoid appears, click “Stop.”
d) To find the period (T), click the box next to “Cursors On” in the bottom left corner of the
oscilloscope window. Using Figure 3.1 as a guide, drag cursor 1 and 2 to positions along the
sinusoid where corresponding x-intercepts exist.
4.
Start with the circuit setup from Figure 3.7. Move the green wire coming from AI 0+ so that it is
no longer connected to either end of the capacitor but only to one end of the inductor. The
circuit should now look like Figure 3.9.
a) NOTE: The resistor and wires do not have to be in the exact configuration that is shown on
the breadboard.
37
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
- +
j i h g f
e d c b a
- +
- +
j i h g f
e d c b a
- +
Figure 3.9: Example Breadboard for Part 2 (b)
5.
NOTE: Figure 3.10 below contains two circuit schematics for Part 2 (a) and (b). The circuit
schematic on the left can be used to represent Figure 3.7 while the schematic on the right can
be used to represent Figure 3.9. Compare the schematics to the figures.
38
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
AO 0
AO 0
0.1 uF
0.1 uF
47 mH
47 mH
AI 0 +
AI 0 +
560 ohms
560 ohms
560 ohms
560 ohms
AGND
AGND
Figure 3.10: Circuit Schematics for Part 2 (a) and (b)
6.
Repeat Steps 2 and 3 for the signal across the inductor (VL) using the circuit from Figure 3.9.
7.
Add your measured values to Table 3.3. Wait to do the calculations.
Table 3.3: Sinusoid Measurements for Part 2
Signal
Vc
VL
Function Generator
Voltage Frequency
(Vp−p )
(Hz)
2
2400
2
2400
Oscilloscope (Measured)
2*A
f
T
(Vp−p )
(Hz)
(sec)
Calculated
ωf
ωT
(rad/sec) (rad/sec)
T
(sec)
39
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
3.4.3 Sinusoidal Measurements – Part 3
Objectives:
 Obtain measurements using an oscilloscope.
o Use capacitors, inductors, and different input/output channels.
 Use a function generator.
o Apply different settings to obtain various sinusoids.
 Discuss the properties of sinusoids.
o Measure amplitude, period, and frequency.
Tasks:
1. Start with the circuit setup from Figure 3.9. Add a yellow wire from AI 1+ so that it is connected
to both the capacitor and inductor along a common connection point. The circuit should now
look similar to Figure 3.11.
- +
j i h g f
e d c b a
- +
- +
j i h g f
e d c b a
- +
Figure 3.11: Example Breadboard for Part 3 (a)
40
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
2.
Apply the same settings to the function generator as those used in Part 2 of the lab procedure
and click “Run.”
3.
Use the oscilloscope to find the peak-to-peak voltage (Vp-p), frequency (f), and time shift (ts) of
the sinusoidal function for the signal across the capacitor (Vc) and complete the corresponding
row of Table 3.4.
a) Set “Timebase” to 200 us (where “us” means micro or 10-6 seconds).
b) Click the “Enabled” box under “Channel 1 Settings.”
c) Set “Scale Volts/Div” to “200mV” under “Channel 0 Settings” and “Channel 1 Settings.”
d) Click “Run.”
e) Once a sinusoid appears, click “Stop.”
f) To find the time shift (ts), click the box next to “Cursors On” in the bottom left corner of the
oscilloscope window. Using Figure 3.1 as a guide, drag cursor 1 (C1 – CH 0) to a position
along the green sinusoid and cursor 2 (C2 – CH 1) to a position along the blue sinusoid where
the gap or offset between the two plots can be determined.
4.
Start with the circuit setup from Figure 3.11. Move the green wire coming from AI 0+ so that it
is no longer connected to either end of the capacitor but only to one end of the inductor. The
circuit should now look like Figure 3.12.
41
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
- +
j i h g f
e d c b a
- +
- +
j i h g f
e d c b a
- +
Figure 3.12: Example Breadboard for Part 3 (b)
5.
NOTE: Figure 3.13 below contains two circuit schematics for Part 3 (a) and (b). The circuit
schematic on the left can be used to represent Figure 3.1 while the schematic on the right can
be used to represent Figure 3.12. Compare the schematics to the figures.
42
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
AO 0
AO 0
AI 1+
AI 1+
0.1 uF
0.1 uF
47 mH
47 mH
AI 0 +
AI 0 +
560 ohms
560 ohms
560 ohms
560 ohms
AGND
AGND
Figure 3.13: Circuit Schematics for Part 3 (a) and (b)
6.
Repeat Steps 2 and 3 for the signal across the inductor (VL) using the circuit from Figure 3.11.
7.
Add your measured values to Table 3.4. Wait to do the calculations.
Table 3.4: Sinusoid Measurements for Part 3
Signal
Vc
VL
Function Generator
Voltage Frequency
(Vp−p )
(Hz)
2
2400
2
2400
Oscilloscope (Measured)
2*ACH0 2*ACH1
ts
(Vp−p )
(Vp−p ) (sec)
Calculated
ϕs
ϕs
(rad) (deg)
43
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
3.4.4 Data Analysis
Sinusoidal Measurements – Part 1
1.
Complete Table 3.2 by performing necessary calculations and attach at the end of the lab.
2.
Attach corresponding hand calculations at the end of the lab.
Sinusoidal Measurements – Part 2
1.
Complete Table 3.3 by performing necessary calculations and attach at the end of the lab.
2.
Attach corresponding hand calculations at the end of the lab.
Sinusoidal Measurements – Part 3
1. Complete Table 3.4 by performing necessary calculations and attach at the end of the lab.
2. Attach corresponding hand calculations at the end of the lab.
3. Write out the equations of the following sinusoids and attach at the end of the lab.
a) Measured
o
Use the values from Table 3.3 and 3.4.
(2∗𝐴)𝐶ℎ0 +(2∗𝐴)𝐶ℎ1
2
o
𝑉𝐶 =
o
𝑉𝐶 (𝑡) =
𝑉𝑐
𝑐𝑜𝑠(𝜔𝑓 𝑡
2
+ ∅𝐶 )
o
𝑉𝐿 (𝑡) =
𝑉𝐿
𝑐𝑜𝑠(𝜔𝑓 𝑡
2
+ ∅𝐿 )
, 𝑉𝐿 =
(2∗𝐴)𝐶ℎ0 +(2∗𝐴)𝐶ℎ1
2
b) Calculated
o
Assume 𝑉𝐶 = 𝑉𝐿 = √2, 𝜔 = 2𝜋𝑓, 𝑓 = 2400 𝐻𝑧, ∅𝐶 = 45° , ∅𝐿 = −45°
o
𝑉𝐶 (𝑡) =
o
𝑉𝐿 (𝑡) =
𝑉𝐶
2
𝑐𝑜𝑠(𝜔𝑡 + ∅𝐶 )
𝑉𝐿
cos(𝜔𝑡
2
+ ∅𝐿 )
c) Through circuit analysis and the above assumptions, the source voltage VS can be
𝑉
represented by, 𝑉𝑆 (𝑡) = 𝑉𝐶 (𝑡) + 𝑉𝐿 (𝑡) = 𝑆𝑃−𝑃 𝑐𝑜𝑠(𝜔𝑡 + ∅).
2
o
Verify this by adding the calculated sinusoid equations together. See “Section 6.5”
(pg. 171) in the Rattan/Klingbeil textbook for more information on adding sinusoids.
44
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
4. Write a MATLAB code to illustrate the relationship between VC , VL , and VS.
o
Plot the calculated equations of VC , VL , and VS on the same plot.



Assume 𝑇 = 0.000417 𝑠𝑒𝑐𝑜𝑛𝑑𝑠.
𝑇
Set the time interval (x-axis) to 𝑡 = (−100 ∗ 𝑇) ∶ (16) ∶ (100 ∗ 𝑇). This
represents the interval starting at -100*T and increasing by T/16 until it
reaches 100*T.
Include the m-file and plot with x and y labels, title, and legend.
45
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
3.5 New MATLAB Commands
hold on

This command allows multiple graphs to be placed on the same X-Y axis and is placed after the
first plot statement.
legend (’string 1’, ’string2’, ‘string3’)

This command adds a legend to the plot. Strings must be placed in the same order as the one in
which the plots were generated.
plot (x, y, ‘line specifiers’)

This command plots the data and uses line specifiers to differentiate between various plots on
the same X-Y axis. In this lab, only use different line styles from the following table.
Table 3.5: Line specifiers for the “plot” command
sqrt(X)

This command produces the square root of the elements of X.
NOTE: The “help” command in MATLAB can be used to find a description and example for functions
such as legend.

For example, type “help legend” in the command window to learn more about the legend
function.
NOTE: Refer to the “MATLAB Commands” sections from prior labs for previously discussed material that
you may also need in order to complete this assignment.
46
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
3.6 General Guidelines (Executive Summary)
1. Write an executive summary for this lab using the Grading Guidelines on the following page.
2. Answer the following questions within the body of the executive summary:
a) Are the measured VC and VL values the same as those calculated? If they are different, list a
few reasons why.
b) Compare the time shifts for VC and VL. Do they differ? If so, how?
3. Place Executive Summary, Appendix and Grading Guidelines in one word document and submit to
Carmen Dropbox. (10 Points will be deducted if the Grading Guidelines sheet is missing)
47
Laboratory 3 Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals
3.7 Sinusoids in Engineering Grading Guidelines (Executive Summary)
Section
Point
Breakdown
Header Information (See Tech. Comm. Guide)
Executive Summary
5
50
Clearly address the purpose of examining the basic properties of sinusoids
and discuss the principles that were addressed throughout the lab.
10
Briefly describe the main aspects of the lab procedure with proper
transitions between each step of the lab. If necessary, reference figures
from the procedure and attach them to the Appendix.
15
Overall, what did each of the tasks teach the team about the basic
properties of sinusoids, the relationship between sinusoid characteristics,
sinusoidal addition, or MATLAB?
15
Incorporate answers to the lab questions within the body of the executive
summary. (See Section 3.6.2)
5
Clearly discuss the obstacles faced throughout the lab and address how
they were overcome. Did the team have problems with the equipment,
taking measurements, etc.?
3
Provide concrete examples of how to improve the lab procedure in order to
improve understanding of basic trigonometric functions.
2
Writing Style
10
Grammar: See Technical Communication Guide for recommendations.
5
Organization and Progression: make sure the summary has natural breaks,
ideas are properly separated by paragraphs and topic sentences, and ideas
are fluently connected.
5
Appendix
35
Tables 3.2, 3.3, and 3.4
9
Hand calculations for Tables 3.2, 3.3 and 3.4 (one example per equation)
12
Sinusoid Equations (measured and calculated)
2
MATLAB m-file for calculated VC(t), VL(t), and VS(t) values
3
MATLAB plots of VC (t), VL (t), and VS (t) for the calculated values
6
Group Grade
Points
Earned
100
48
Laboratory 4 – Systems of Equations in Engineering: The TwoLoop Circuit
4.1 Laboratory Objective
The objective of this laboratory is to learn the basics of systems of equations, matrices, and their
application in engineering.
4.2 Educational Objective
After performing this experiment, students should be able to:
1. Identify and create a two-loop circuit.
2. Solve for unknown electrical values by use of a matrix inverse, Cramer’s Rule, substitution, and
MATLAB.
3. Apply new functions in MATLAB toward the solution of engineering equations.
49
Laboratory 4 Systems of Equations in Engineering: The Two-Loop Circuit
4.3 Background
Simultaneous equation solving is a key skill in many engineering applications. For example, software in
finite element modeling and thermodynamics solve multiple equations with multiple variables. MATLAB
is one such piece of software that can solve simultaneous equations. The following methods will solve
relatively small problems easily and can provide an answer quickly, however with larger systems, hand
calculations can be exhaustive and software becomes the more intelligent route to the solution.
4.3.1 Problem Statement
A system of equations can be written as 𝐴𝑥⃗ = 𝑏⃗⃗, where A is the coefficient matrix, 𝑥⃗ is a vector of
unknowns, and 𝑏⃗⃗ is a vector of the right hand sides of the equations. For illustration, the following
matrices will be used for explaining the methods.
𝑎
𝐴=[
𝑐
𝑥
𝑥⃗ = [𝑦]
𝑏
]
𝑑
𝑚
𝑏⃗⃗ = [ ]
𝑛
4.3.2 Matrix Inverse Method
𝑥⃗ = 𝐴−1 𝑏⃗⃗
𝐴−1 =
1
𝑎
|
𝑐
𝑏
|
𝑑
𝑑
[
−𝑐
−𝑏
]
𝑎
(4.1)
4.3.3 Cramer’s Rule
𝑥=
𝑚 𝑏
|
𝑛 𝑑
𝑎 𝑏
|
|
𝑐 𝑑
|
𝑦=
𝑎 𝑚
|
𝑐 𝑛
𝑎 𝑏
|
|
𝑐 𝑑
|
4.3.4 Substitution
The system 𝐴𝑥⃗ = 𝑏⃗⃗ can be written as two equations.
𝑎𝑥 + 𝑏𝑦 = 𝑚
𝑐𝑥 + 𝑑𝑦 = 𝑛
(4.2)
(4.3)
Solve for x in Equation 4.3 and substitute into Equation 4.2. Once y is known, substitute back into either
Equation 4.2 or 4.3, and solve for x.
50
Laboratory 4 Systems of Equations in Engineering: The Two-Loop Circuit
4.3.5 MATLAB
MATLAB will solve the problem by two methods.
𝑥⃗ = 𝐴−1 𝑏⃗⃗
𝑥⃗ = 𝐴 \ 𝑏⃗⃗
NOTE: The second method is pronounced “A left division b” and is much more efficient for larger
matrices than method one.
51
Laboratory 4 Systems of Equations in Engineering: The Two-Loop Circuit
4.4 Procedure
4.4.1 Two-Loop Circuit
Objectives:
 Identify and create a two-loop circuit
o Use resistors, ammeters and a voltage supply
 Solve for unknown electrical values by use of a matrix inverse, Cramer’s Rule, substitution, and
MATLAB
 Apply new functions in MATLAB toward the solution of engineering equations
o Verify the measured current
Tasks:
Follow the steps outlined below after the Lab Teaching Assistant has explained how to use the
laboratory equipment.
1. The currents flowing through loop 1 and loop 2 in the circuit below are unknown. Construct the
circuit.
 NOTE: R1 = R2 = 100 Ω and R3 = R4 = 200 Ω.
R3
R1
+
VS
Loop1
Loop2
R2
Ammeter
−
R4
Figure 4.1: A Two-Loop Circuit
2. Use the lab equipment to find I2, the current in loop 2.
3. Record the measured values in Table 4.1.
Table 4.1 Two-Loop Circuit
VS (Volts)
I2 (Amps)
I1 (Amps)
5
7
52
Laboratory 4 Systems of Equations in Engineering: The Two-Loop Circuit
4.4.2 Data Analysis
1.
Calculate I1, the current in loop 1, and record the value in Table 4.1. Attach the completed table at
the end of the lab.
2.
By hand, calculate the unknown currents, I1 and I2 using the three methods below and attach at
the end of the lab.
a) Inverse Matrix Method
b) Cramer’s Rule
c) Substitution
o NOTE: Use the following matrix setup (Take VS to be 7):
[
3.
(𝑅1 + 𝑅2 )
−𝑅2
𝐼
𝑉
] [ 1] = [ 𝑆]
−𝑅2
(𝑅2 + 𝑅3 + 𝑅4 ) 𝐼2
0
Write a MATLAB code implementing the Inverse Matrix Method. Have the user input values for all
resistors and the voltage source. Check your answer with “Left Division.” At the end of the lab,
attach: (both cases where VS = 5 and 7 V)
a) m-file
b) output
53
Laboratory 4 Systems of Equations in Engineering: The Two-Loop Circuit
4.5 New MATLAB Commands
x = input (‘text’);

The “input” command displays the ‘text’ inside of the parentheses as a prompt to the user.
The user then responds to the prompt by typing a value which is stored under the chosen
variable name ‘x.’
NOTE: A dot operator is not necessary for multiplication, division, and exponentiation between the
matrixes since you have a 2-by-2 matrix and 2-by-1 matrix.
NOTE: The “help” command in MATLAB can be used to find a description and example for functions
such as input.

For example, type “help input” in the command window to learn more about the input
function.
NOTE: Refer to the “MATLAB Commands” sections from prior labs for previously discussed material that
you may also need in order to complete this assignment.
54
Laboratory 4 Systems of Equations in Engineering: The Two-Loop Circuit
4.6 General Guidelines (Executive Summary)
1. Write an executive summary for this lab using the Grading Guidelines on the following page.
2. Answer the following questions within the body of the executive summary:
a) Compare your calculated values for I1 and I2 with the measured values. Why are they
different?
3. Place Executive Summary, Appendix, and Grading Guidelines in one Word document and submit to
Carmen Dropbox. (10 Points will be deducted if the Grading Guidelines sheet is missing)
55
Laboratory 4 Systems of Equations in Engineering: The Two-Loop Circuit
4.7 Systems of Equations Grading Guidelines (Executive Summary)
Section
Point
Breakdown
Header Information (See Tech. Comm. Guide)
Executive Summary
5
55
Clearly address the purpose of examining the two-loop circuit and discuss
the principles that were addressed throughout the lab.
10
Describe the main aspects of the lab procedure with proper transitions
between each step of the lab.
15
Overall, what did each of the tasks teach the team about matrices, solving
multiple equations simultaneously, or MATLAB?
20
Incorporate answers to the lab questions within the body of the executive
summary. (See Section 4.5.2)
5
Clearly discuss the obstacles faced throughout the lab and address how
they were overcome. Did the team have problems with the equipment,
ammeter, etc.?
3
Provide concrete examples of how to improve the lab procedure in order to
improve understanding of matrices.
2
Writing Style
10
Grammar: See Technical Communication Guide for recommendations.
5
Organization and Progression: make sure the summary has natural breaks,
ideas are properly separated by paragraphs and topic sentences, and ideas
are fluently connected.
5
Appendix
30
Table 4.1
2
Hand calculations using Inverse, Cramer’s Rule, and Substitution
(one example per equation)
15
MATLAB m-file
5
MATLAB output
5
Lab Participation Agreement
3
Group Grade
Points
Earned
100
56
Laboratory 5 Derivatives in Engineering: Velocity and Acceleration in Free-Fall
Laboratory 5 – Derivatives in Engineering: Velocity and
Acceleration in Free-Fall
5.1 Laboratory Objective
The objective of this laboratory is to illustrate the application of a derivative with a free-fall exercise.
5.2 Educational Objective
After performing this experiment, students should be able to:
1. Understand the relationship between position, velocity, and acceleration.
2. Identify the key parameters of free-fall.
3. Understand the importance of data accuracy.
57
Laboratory 5 Derivatives in Engineering: Velocity and Acceleration in Free-Fall
5.3 Background
The derivative is a tool that describes the rate of change of a quantity with respect to the change in
another. Geometrically this is equivalent to slope.
Given a function 𝑦(𝑡) that represents position with respect to time, one can derive the expressions for
the velocity 𝑣(𝑡) and the acceleration 𝑎(𝑡). Velocity is simply the derivative of 𝑦(𝑡) with respect to time
and acceleration is the second derivative of 𝑦(𝑡) with respect to time.
𝑣(𝑡) =
𝑑𝑦
𝑑𝑡
𝑑2 𝑦
𝑑𝑣
𝑎(𝑡) =
=
𝑑𝑡 2
𝑑𝑡
Velocity can also be calculated using
Δ𝑦
,
Δ𝑡
or
𝑣(𝑡) =
𝑦2 −𝑦1
𝑡2 − 𝑡1
(5.1)
𝑣2 −𝑣1
𝑡2 − 𝑡1
(5.2)
Δ𝑣
Similarly, acceleration can be calculated using Δ𝑡 , or
𝑎(𝑡) =
An object in free-fall that is dropped from an elevated height has a constant acceleration of g = -9.81
m/s2 and an initial velocity, v0 = 0 m/s. For these free-fall objects, the following kinematic equations of
motion can be used.
1
𝑥 = 𝑥0 + 𝑣0 ∗ ∆𝑡 + 2 ∗ 𝑎 ∗ (∆𝑡)2
(5.3)
𝑣 = 𝑣0 + 𝑎 ∗ ∆𝑡
(5.4)
𝑣 2 = 𝑣0 2 + 2 ∗ 𝑎 ∗ ∆𝑥
(5.5)
The apparatus used in this lab consists of a free-fall device, break-beam sensors that are mounted to
fixed positions, and a computer to record the data.
58
Laboratory 5 Derivatives in Engineering: Velocity and Acceleration in Free-Fall
5.4 Procedure
The following apparatus includes a set of 8 evenly spaced break-beam sensors starting at the 7 foot
mark and going to 0 feet. Each break-beam is spaced 1 foot apart, or 0.3048 meters (See Figure 5.1).
Ceiling grid hook
Pin (with ball on top)
String
7 ft.
1 ft.
Break-beam sensor
Foam
Figure 5.1: Free-Fall apparatus with break-beams
Follow the steps outlined below after the Lab Teaching Assistant has explained how to use the
laboratory equipment.
1. Prepare the apparatus for the first drop by making sure foam is attached to the bottom end of the
tube, a pin is in place at the top of the tube, and a ball is sitting on the pin. Hang the tube from the
ceiling grid hook. Make sure the string is not tangled at the point where it is attached to the pin.
2. Download and save the following two m-files from the course website to a convenient location on
the computer:
 dropball.m
 dropballDataAnalysis.m
3. Open MATLAB and make sure your current directory is set to the location which contains the above
two m-files.
59
Laboratory 5 Derivatives in Engineering: Velocity and Acceleration in Free-Fall
4. Connect the data acquisition (DAQ) board to the USB port of the computer. The breadboard will be
pre-wired. Make sure the tube is hooked to the DAQ breadboard using the telephone cable. Wait
for the tube to “light up” and for the following messages to appear in the bottom right corner of
your computer screen “NI myDAQ detected!”
5. Run dropball.m once to make sure the DAQ has been identified but do not drop the ball. You should
get a result similar to Figure 5.2.
Figure 5.2: MATLAB output before dropping ball

The above graph is showing 10 seconds of noise (35 millivolts) on the 8 break-beam sensors
that are connected in parallel.
5.4.1 Sample Rate #1 (Default Settings)
Objectives:
 Understand the relationship between position, velocity, and acceleration.
o Calculate acceleration and velocity using position.
 Identify the key parameters of free-fall.
Tasks:
1.
Now you’re ready to drop the ball. The default settings are:
 Frequency = 20,000 samples/sec or Hz.
 10 second capture window.
 Data will be saved to a file called ‘dropballData.mat.’
 NOTE: These settings can be changed at the top of the dropball.m script in the section,
which looks like the following one,
60
Laboratory 5 Derivatives in Engineering: Velocity and Acceleration in Free-Fall
%% Set sample rate and data rate here (please do not modify any other code)
% ------------------------------------------------------------------------%
the default sample rate is 20000 per sec
sr = 20000;
%
the default collection window length is 10 seconds
sd = 10;
%
the matlab (mat) file for data storage
%
this is the file that will be analyzed using dropballAnalysis.m
filename='dropballData';
2. You will be doing this test with two additional sample rates later in the lab. So, the .mat file should
be given a unique name for each sample rate. Change line 24 of dropball.m using the following
example.
%
the matlab (mat) file for data storage
%
this is the file that will be analyzed using dropballAnalysis.m
filename='dropballData20000';
3. Make sure dropball.m is in your workspace window or on your MATLAB path. Click the green “Run”
button in the dropball.m MATLAB script file or type “run dropball.m” in the Command Window.
Wait for the following words to appear in the Command Window.
Please drop the ball! You have a 10 second window!
4. Now, pull the pin within 10 seconds. After a bit of a wait (we’re moving 2 times 20,000 data points),
you should get a graph that looks like Figure 5.3. However, your graph may be shifted along the xaxis depending on when you pull the pin.
Figure 5.3: MATLAB output after dropping ball (Default Settings)

NOTE: the voltage ranges from 0.5 volts to 4 volts. Each four-volt spike is the result of a
break-beam being broken by the one inch ball during the fall.
61
Laboratory 5 Derivatives in Engineering: Velocity and Acceleration in Free-Fall
5. Once data is collected, it can be further analyzed by running the dropballDataAnalysis.m. Make
sure you modify the code to match the filename from Step 2. See line 9 of the
dropballDataAnalysis.m script file, which looks similar to the following code in which “20000”
represents sampling rate #1.
%% load data taken using the dropball script
load('dropballData20000');
6. When the run is complete, you will be given a two-frame subplot for the 8 break-beam pulses.
Attach the two-frame subplot for each sample rate in the Appendix. You’ll also get a comma
separated distance and time for each of the eight leading edges of the ball drop through the breakbeams. Create vectors in MATLAB containing the distance and time data.
Figure 5.4: MATLAB output after dropping and analyzing ball (Default Settings)
5.4.2 Sample Rate #2 (Open-Source Arduino Settings)
Objectives:
 Understand the relationship between position, velocity, and acceleration.
o Calculate acceleration, and velocity using position.
 Identify the key parameters of free-fall.
 Understand the importance of data accuracy.
o Is the sampling rate sufficient?
62
Laboratory 5 Derivatives in Engineering: Velocity and Acceleration in Free-Fall
Tasks:
Inexpensive microprocessors such as the open-source Arduino are readily available these days. The
Arduino can easily sample at a rate of 2,000 samples per second. It is helpful to determine if the
sampling rate is sufficient to obtain usable data for this experiment.
1. Run the experiment at 2,000 samples per second and then repeat Steps 2-6 from section “5.4.1
Sample Rate #1 (Default Settings)” to verify the results (or lack of results).
5.4.3 Sample Rate #3 (MATLAB Arduino Settings)
Objectives:
 Understand the relationship between position, velocity, and acceleration.
o Calculate acceleration and velocity using position.
 Identify the key parameters of free-fall.
 Understand the importance of data accuracy.
o Is the sampling rate sufficient?
Tasks:
The Arduino can communicate directly with MATLAB using free ArduinoIO software found on MATLAB
Central. Unfortunately, the communication is done using the serial bus which is relatively slow. In this
setup, the Arduino can only sample at a rate of 200 samples per second. Is that sampling rate sufficient
to obtain usable data for this experiment?
1. Run the experiment at 200 samples per second and then repeat Steps 2-6 from section “5.4.1
Sample Rate #1 (Default Settings)” to verify the results (or lack of results).
5.4.4 Data Analysis
Sample Rate #1 (Default Settings)
1. Using MATLAB and MS Word, construct Table 5.1. Plot the measured position vs. time, velocity vs.
time, and acceleration vs. time. Make sure you add appropriate title and axis labels. You will only
insert the plots for sample rate #1 in your Appendix.
63
Laboratory 5 Derivatives in Engineering: Velocity and Acceleration in Free-Fall
Table 5.1: Position, Velocity, and Acceleration
𝑡𝑖 (𝑠)
𝑦𝑖 (𝑚)
∆𝑡 = 𝑡𝑖+1
− 𝑡𝑖 (𝑠)
∆𝑦
= |𝑦𝑖+1 − 𝑦𝑖 |(𝑚)
0
t1
t2
etc.
0
y1
y2
etc.
t1 −t0
t2 −t1
etc.
y1 −y0
y2 −y1
etc.
𝑣(𝑡) =
Δ𝑦 𝑚
( )
Δ𝑡 s
𝑚
∆𝑣 = |𝑣𝑖+1 − 𝑣𝑖 | ( )
𝑠
0
v1
v2
etc.
v1 −v0
v2 −v1
etc.
𝑎(𝑡) =
Δ𝑣 𝑚
( )
Δ𝑡 s 2
0
a1
a2
etc.
Sample Rate #2 (Open-Source Arduino Settings)
1. Reproduce the above table for Sample Rate #2. However, label it as Table 5.2 and adjust the titles
on the position, velocity, and acceleration plots.
2. Determine if the current sampling rate is sufficient to obtain usable data for this experiment.
 Hints:
1
2000
o
For a sampling rate or frequency of 2,000, ∆𝑡 =
o
o
Refer to equations 5.3-5.5 in section “5.3 Background.”
For our experiment, maximum velocity is obtained at 7 feet or 2.1336 meters. The ball
is one inch or 0.0254 meters in diameter.
Figure out how long it takes the ball to travel one inch with a velocity obtained at the
seven foot mark.
 First, using equation 5.5, find the maximum velocity which occurs at the seven
foot mark.
 Next, to determine the time required for the ball to pass the break-beam
sensor, divide the diameter of the ball by the maximum velocity.
 Invert the time required for the ball to pass the break-beam sensor to
determine the highest frequency of samples being measured.
The Nyquist rate is the minimum rate at which a signal can be sampled without
introducing errors. It is twice the highest frequency of samples being measured in the
signal. Calculate the Nyquist rate using the above frequency.
Compare the Nyquist rate with the sampling rate. Is the sampling rate sufficient to
obtain usable data for this experiment?
o
o
o
seconds.
Sample Rate #3 (MATLAB Arduino Settings)
1. Reproduce the above table for Sample Rate #3. However, label it as Table 5.3 and adjust the
titles on the position, velocity, and acceleration plots.
2. Use your calculations from Sample Rate #2 to determine if this sampling rate is sufficient.
64
Laboratory 5 Derivatives in Engineering: Velocity and Acceleration in Free-Fall
5.5 General Guidelines (Executive Summary)
1. Write an Executive Summary for this lab using the Grading Guidelines on the following page.
2. Answer the following questions within the body of the executive summary:
a) Compare the two-frame subplots for sample rate number 1 through 3. Are there noticeable
differences between them?
b) In free-fall, what physical quantity does the acceleration represent?
c) What is the mathematical relationship between position, velocity, and acceleration?
3. Place Executive Summary, Appendix and Grading Guidelines in one word document and submit to
Carmen Dropbox. (10 Points will be deducted if the Grading Guidelines sheet is missing)
65
Laboratory 5 Derivatives in Engineering: Velocity and Acceleration in Free-Fall
5.6 Derivatives in Engineering Grading Guidelines (Executive Summary)
Section
Point
Breakdown
Header Information (See Tech. Comm. Guide)
Executive Summary
5
55
Clearly address the purpose of the free-fall exercise and discuss the principles
that were addressed throughout the lab.
10
Describe the main aspects of the lab procedure with proper transitions
between each step of the lab.
15
Overall, what did each of the tasks teach the team about derivatives, position,
velocity, acceleration, sampling rate, etc.?
15
Incorporate answers to the lab questions within the body of the executive
summary. (See Section 5.5.3)
10
Clearly discuss the obstacles faced throughout the lab and address how they
were overcome. Did the team have problems with the equipment, MATLAB,
etc.?
3
Provide concrete examples of how to improve the lab procedure in order to
improve understanding of derivatives.
2
Writing Style
10
Grammar: See Technical Communication Guide for recommendations.
5
Organization and Progression: make sure the summary has natural breaks,
ideas are properly separated by paragraphs and topic sentences, and ideas
are fluently connected.
5
Appendix
30
Table 5.1. 5.2, 5.3
9
MATLAB two-frame subplot for sample rate 1, 2, 3
6
Excel or Word position, velocity & acceleration plots for 20,000 Hz
6
Hand Calculations to determine if sampling rate was sufficient
6
Lab Participation Agreement
3
Group Grade
Points
Earned
100
66
Laboratory 6 Integrals in Engineering: Work and Stored Energy in a Spring
Laboratory 6 – Integrals in Engineering: Work and Stored
Energy in a Spring
6.1 Laboratory Objective
The objective of this laboratory is to illustrate the application of an integral through an exercise with
spring work.
6.2 Educational Objective
After performing this experiment, students should be able to:
1. Understand that geometrically an integral calculates the area under a curve.
2. Understand the work done on a spring.
67
Laboratory 6 Integrals in Engineering: Work and Stored Energy in a Spring
6.3 Background
The integral of a function is the area under the function. For an example, see Figure 9.8 on page 300 of
Rattan/Klingbeil textbook.
Work is a fundamental concept of many physical systems. In general, the sum of all forces over a given
distance is work.
𝑥
𝑥
𝑊 = ∫𝑥 𝑖 𝑓(𝑥) 𝑑𝑥 = [𝐹(𝑥)]𝑥0𝑖 = 𝐹(𝑥𝑖 ) − 𝐹(𝑥0 )
0
(6.1)
A spring has work done on it when it is stretched. The spring force is linearly related to the distance
stretched by a constant, k.
𝐹𝑜𝑟𝑐𝑒(𝑥) = 𝑘𝑥
(6.2)
The work done on a spring by a mass can be found by substituting Equation 6.2 into Equation 6.1.
𝑥
𝑊 = ∫𝑥 𝑖 𝑘𝑥 𝑑𝑥
0
(6.3)
Figure 6.1 below shows the setup that will be used in the lab.
46 cm
Figure 6.1: Spring & Mass System
68
Laboratory 6 Integrals in Engineering: Work and Stored Energy in a Spring
6.4 Procedure
6.4.1 Work and Stored Energy in a Spring
Objectives:
 Understand that geometrically an integral calculates the area under a curve.
o Verify this with different plots.
 Understand the work done on a spring.
Tasks:
Follow the steps outlined below after the Lab Teaching Assistant has explained how to use the
laboratory equipment.
1. Attach the spring to the stand and suspend the mass hanger from the other end. Add one weight
(0.048 kg) to the mass hanger.
2. Record the measurement from the scale in Table 6.1. This is your reference value xo.
3. Add mass (1 weight at a time) as shown in Table 6.1 and record the measurements.
Table 6.1: Measurements with Weights/Masses
# of Weights
1
2
3
4
5
6
7
8
Mass (kg)
0.048
0.096
0.144
0.192
0.240
0.288
0.336
0.384
Position
x0
x1
x2
x3
x4
x5
x6
x7
Height (m)
∆x = |xi -x0| (m)
-
base = |xi -x(i-1)|
-
F (N)
-
6.4.2 Data Analysis
1. Complete Table 6.1.
 Use the following equation to calculate the force on the spring in Newtons. (g = 9.81 m/s2)
𝐹 = 𝑚𝑔
 Calculate ∆x due to each added mass.
69
Laboratory 6 Integrals in Engineering: Work and Stored Energy in a Spring
o
NOTE: Remember to take the absolute value of your answers in columns five and six.
Also, be sure to convert your height measurements from cm to m.
2. Using Excel, plot Force vs. ∆x as a scatterplot. Make sure you include axis labels and a title.
3. Add a trend line to your Excel plot to find the slope and y-intercept of the graph.
4. Copy and paste your plot into another area of the same worksheet. Within the “Design” tab, under
“Chart Tools,” change your chart type to “Area.” Remove the equation of the trend line from the
plot. This is the area you need to find in order to determine the work done on the spring.
5. Calculate the area under the line. This can be done by breaking the area into two simple shapes
and then adding the area of the shapes together. Include hand calculations in Appendix.
 NOTE: You want the area between ∆x1 and ∆x7 .
6. Plot Force vs. Position in Excel as a “Column Chart”.
 When you create the plot, only select the “F(N)” column as your data.
 In the “Layout” tab, select “Data Labels” to display your values.
 Remove your horizontal gridlines, if needed, to see the data labels more clearly.
 Click on the columns of your plot. Then, right click and select “Format Data Series…”.
 In “Series Options,” move the “Gap Width” slider to “No Gap”
 In “Border Color” select “Solid line” and make sure the color is something that
will appear on your plot.
 Your Force vs. Position plot should have a similar shape to your Force vs. ∆x plot. The
column graph approximately represents the area under the curve, though the x values differ
from the Force vs. ∆x plot.
7. Assuming that the width of each column is the base value in your Force vs. Position plot, calculate
the area under the curve. Include hand calculations in Appendix.
8. Calculate the integral for the work done on the spring by hand.
 NOTE: Using equation 6.1, substitute your trend line equation (from the plot of Force vs. Δx)
in place of f(x). The limits of integration are ∆x1 to ∆x7 from Table 6.1.
70
Laboratory 6 Integrals in Engineering: Work and Stored Energy in a Spring
6.5 General Guidelines (Executive Summary)
1. Write an Executive Summary for this lab using the Grading Guidelines on the following page.
2. Answer the following questions within the body of the executive summary:
a) What does the slope of the linear plots physically represent?
b) What are the units of the spring constant k?
c) What does the area of the bar graph physically represent?
d) How does the area of the bar graph compare to the area of the Force vs. Δx plot.
3. Place Executive Summary, Appendix and Grading Guidelines in one word document and submit to
Carmen Dropbox. (10 Points will be deducted if the Grading Guidelines sheet is missing)
71
Laboratory 6 Integrals in Engineering: Work and Stored Energy in a Spring
6.6 Integrals in Engineering Grading Guidelines (Executive Summary)
Section
Point
Breakdown
Header Information (See Tech. Comm. Guide)
Executive Summary
5
55
Clearly address the purpose of examining the spring and discuss the
principles that were addressed throughout the lab.
12
Describe the main aspects of the lab procedure with proper transitions
between each step of the lab.
18
Overall what did each of the tasks teach the team about integrals, work,
force, etc.?
12
Incorporate answers to the lab questions within the body of the executive
summary. (See Section 6.5.3)
8
Clearly discuss the obstacles faced throughout the lab and address how
they were overcome. Did the team have problems with the equipment,
Excel, etc.?
3
Provide concrete examples of how to improve the lab procedure in order to
improve understanding of integrals.
2
Writing Style
10
Grammar: See Technical Communication Guide for recommendations.
5
Organization and Progression: make sure the summary has natural breaks,
ideas are properly separated by paragraphs and topic sentences, and ideas
are fluently connected.
5
Appendix
30
Table 6.1
9
Force vs. Δx Scatterplot, Force vs. Δx Area plot, Force vs. Position plot
6
Hand Calculations for Force vs. Δx plot
3
Hand Calculations for Force vs. Position plot
3
Hand Calculations for integral of the work done on the spring
6
Lab Participation Agreement
3
Group Grade
Points
Earned
100
72
Laboratory 7 – Differential Equations in Engineering: The
Leaking Bucket
7.1 Laboratory Objective
The objective of this laboratory is to learn about first-order differential equations and their application
to a leaking bucket system.
7.2 Educational Objectives
After performing this experiment, students should be able to:
1. Understand the modeling of a leaking bucket dynamic system.
2. Measure the key parameters of a leaking bucket dynamic system.
3. Use observed data to validate a mathematical model of a leaking bucket.
73
Laboratory 7 Differential Equations in Engineering: The Leaking Bucket
7.3 Background
Differential equations are an integral part of engineering. Almost all system response can be described
by a differential equation. Knowing how to solve these problems is a key to one’s success in
engineering. This lab looks at one classification of a differential equation; first-order, constant
coefficient, and homogeneous.
The system shown in Figure 7.1 can be described by investigating the behavior of water flowing into and
out of a bucket.
Top View
Front View
Atank
h(t)
Ahole
Q out
Figure 7.1: Leaking Bucket
The following equation describes the volumetric flow rate, Q of the system.
𝑄𝑖𝑛 − 𝑄𝑜𝑢𝑡 − 𝑄𝑠𝑡𝑜𝑟𝑒𝑑 = 0
Assume the tank is cylindrical in shape and the area is represented by 𝐴𝑡𝑎𝑛𝑘 . Assume the hole is circular
in shape and the area is represented by 𝐴ℎ𝑜𝑙𝑒 .
There will not be any water flowing into our system, therefore 𝑄𝑖𝑛 = 0.
𝑄𝑠𝑡𝑜𝑟𝑒𝑑 = −𝑄𝑜𝑢𝑡
The volumetric flow rate is found by multiplying the velocity by the area.
𝐴𝑡𝑎𝑛𝑘 ℎ̇ = −𝐴ℎ𝑜𝑙𝑒 𝑣
From the study of fluid dynamics, the velocity of water coming out of the straw is √2𝑔ℎ.
𝐴𝑡𝑎𝑛𝑘 ℎ̇ = −𝐴ℎ𝑜𝑙𝑒 √2𝑔ℎ
74
Laboratory 7 Differential Equations in Engineering: The Leaking Bucket
Rearranging terms and writing all constants as one:
𝐴𝑡𝑎𝑛𝑘 ℎ̇ + 𝐴ℎ𝑜𝑙𝑒 √2𝑔ℎ = 0
𝐴𝑡𝑎𝑛𝑘 ℎ̇ + 𝐾√ℎ = 0
The above equation cannot be solved using the methods of this class because the h on the second term
of the equation is under a square root. To accommodate this, the governing equation that will be
solved in this lab will be approximated without the square root:
𝐴𝑡𝑎𝑛𝑘 ℎ̇ + 𝐾ℎ = 0
The solution to the governing equation is:
ℎ(𝑡) = 𝐶𝑒 −(𝐾/𝐴
𝑡𝑎𝑛𝑘 )𝑡
(7.1)
If 𝑄𝑖𝑛 = 𝐵 (in cubic inches per second), where B is a constant, then the governing equation becomes:
ℎ(𝑡) = 𝐶𝑒
−(
𝐾
𝐴𝑡𝑎𝑛𝑘
)𝑡
+
𝐵
𝐾
Where C is the initial height of the water and the system time constant is defined as 𝜏 = 𝐴𝑡𝑎𝑛𝑘 /𝐾.
75
Laboratory 7 Differential Equations in Engineering: The Leaking Bucket
7.4 Procedure
7.4.1 Leaking Bucket System
Objectives:
 Understand the modeling of a leaking bucket dynamic system.
o Recognize the importance of the area of the bucket and hole along with the input and
output flow.
 Measure the key parameters of a leaking bucket dynamic system.
 Validate a mathematical model of the leaking bucket with observed data.
o Use differential equations.
Tasks:
Follow the steps outlined below after the Lab Teaching Assistant has explained how to use the
laboratory equipment.
1. Make sure the power supply for the leaking bucket system is plugged into an outlet.
 NOTE: The system operates on 12 DC Volts.
2. Make sure the clear bucket is filled to a level of approximately 5.5 inches.
 NOTE: If the bucket is not already at this level, with the valve closed, press the silver
pushbutton, which contains a blue circular light.
o The bucket has a diameter of 6 inches and the hole in the bucket is 0.19 inches.
3. Open MATLAB and then open the leakingBucket.m script file. Data will be stored in the variables h
(height in inches) and t (time in seconds).
 Make sure your current directory in MATLAB is set to the computer location that
contains the “m file.”
4. Click on the “Run” button in the leakingBucket.m file window. The following message will appear
in the Command Window:
 Press ENTER key when height reaches 5.0:
5. Open the valve on the leaking bucket system (The handle on the valve swings back toward the
bottle).
6. Press “Enter” when the water level reaches 5 inches. The following message will now appear in
the Command Window:
 Press ENTER key when height reaches 4.5:
7. Again, press “Enter” when the water level reaches the approximate mark. Continue to follow the
prompts in the Command Window until the water level reaches 0 inches.
8. At 0 inches, wait until the nozzle is slowly dripping. This may take 1 – 2 minutes.
9. Once “Enter” has been pressed at a water level of 0 inches, the program will ask if you want to
collect another set of data.
76
Laboratory 7 Differential Equations in Engineering: The Leaking Bucket
 This option is built in so you can compare a run with Qin (flow in) equal to zero, and Qin
equal to 1.05 ml/sec.
 In order to obtain data when Qin = 1.05 ml/sec, type ‘y,’ fill the bottle to 5.5 inches and
turn on the constant flow rate peristaltic pump, using the toggle switch.
o The peristaltic pump will produce a constant flow rate of 1.05 ml/sec of water.
o Follow through with the same procedure, but realize that the output flow will
never be zero.
o Once the water level hits zero you can turn off the pump and wait for it to drip
similarly to the first procedure.
10. Data is displayed in the Command Window and saved in both a ‘.mat’ and ‘.xls’ file.
7.4.2 Data Analysis
1.
If you have collected multiple data sets, you can plot a single data set using the following
notation:

plot(t(:,1),h(:,1),'*-') -> [for the first data set]

grid ON

plot(t(:,2),h(:,2),'*-') -> [for the second data set and so forth]

grid ON
2.
Once a plot of height versus time has been produced for your data sets, create a plot of log(h) vs
time. Since log(0) is infinite, plot the data sets as follows:

plot(t(1:end-1),log(h(1:end-1)))
-> [if only one data set is recorded]

grid ON

plot(t(1:end-1,1),log(h(1:end-1,1))) -> [if using multiple data sets]

grid ON
3.
Be sure to include axis labels and a title on all plots. Also, for the log plot, add a linear line and
equation (See section “1.5 MATLAB Commands” for more details.)
4.
Using the ‘.xls’ file, “myData”, complete Table 7.1 below for the leaking bucket system.
77
Laboratory 7 Differential Equations in Engineering: The Leaking Bucket
Table 7.1: Leaking Bucket System
Trial 1
Height (inches) Time (seconds)
5
0
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
Trial 2
ΔT
(T2-T1)
(T3-T2)
(T4-T3)
Time(seconds)
0
0
ΔT
0
(T2-T1)
(T3-T2)
(T4-T3)
5. Derive by hand the equation of the straight line for the “ln plot” in terms of C, K, and Atank when Qin
is equal to zero.
 HINT: Start by taking the natural log of both sides of Equation 7.1 and algebraically
simplify
78
Laboratory 7 Differential Equations in Engineering: The Leaking Bucket
7.5 General Guidelines (Executive Summary)
1. Write an Executive Summary for this lab using the Grading Guidelines on the following page.
2. Answer the following questions within the body of the executive summary:
a) What is the time constant, τ? (Don’t forget units.)
b) What type of energy is stored in the water?
c) Compare the plots and ΔT when Qin is zero and when Qin is 1/05 ml/sec.
3. Place Executive Summary, Appendix and Grading Guidelines in one word document and submit to
Carmen Dropbox. (10 Points will be deducted if the Grading Guidelines sheet is missing)
79
Laboratory 7 Differential Equations in Engineering: The Leaking Bucket
7.6 Differential Equations in Engineering Grading Guidelines (Executive
Summary)
Section
Point
Breakdown
Header Information (See Tech. Comm. Guide)
Executive Summary
5
65
Clearly address the purpose of examining the leaking bucket system and
discuss the principles that were addressed throughout the lab.
15
Describe the main aspects of the lab procedure with proper transitions
between each step of the lab.
20
Overall, what did each of the tasks teach the team about differential
equations, flow rates, etc.
15
Incorporate answers to the lab questions within the body of the executive
summary. (See Section 7.5.3)
10
Clearly discuss the obstacles faced throughout the lab and address how
they were overcome. Did the team have problems with the equipment,
MATLAB, etc.?
3
Provide concrete examples of how to improve the lab procedure in order to
increase understanding of differential equations.
2
Writing Style
10
Grammar: See Technical Communication Guide for recommendations.
5
Organization and Progression: make sure the summary has natural breaks,
ideas are properly separated by paragraphs and topic sentences, and ideas
are fluently connected.
5
Appendix
20
Table 7.1
2
Height vs. time plots, log(height) vs. time plots
9
Hand Calculations for equation 7.1
6
Lab Participation Agreement
3
Group Grade
Points
Earned
100
80
Laboratory 8 – Differential Equations in Engineering: SpringMass Vibration
8.1 Laboratory Objective
The objective of this laboratory is to model spring-mass behavior with a second-order differential
equation.
8.2 Educational Objective
After performing this experiment, students should be able to:
1. Apply principles of modeling and analysis to a spring-mass system.
2. Identify and measure the key parameters of a spring-mass system.
3. Validate a mathematical model (differential equation) with measured data.
81
Laboratory 8 Differential Equations in Engineering: Spring-Mass Vibration
8.3 Background
Another class of differential equations are second-order applications. These equations contain a second
derivative of the variable in question. In the case of a spring-mass system, the displacement as a
function of time is the unknown quantity.
The spring-mass system shown in Figure 8.1 has kinetic energy associated with the mass moving up and
down and potential energy stored in the spring. This energy is passed back and forth as the spring
oscillates. The equation y(t) is the position measured from equilibrium.
K
M
y(t)
Figure 8.1: Spring-Mass System
The free body diagram (FBD) in Figure 8.2 shows all forces acting on the mass.
k𝛿
mg
Figure 8.2: Free Body Diagram of Spring-Mass System
From equilibrium of forces in the y-direction, where k is the spring constant, 𝛿 is the equilibrium
𝑚
elongation of the spring, m is the mass and g is gravity or 9.81 𝑠2 , the following equation is produced:
82
Laboratory 8 Differential Equations in Engineering: Spring-Mass Vibration
𝑘𝛿 = 𝑚𝑔,
which gives,
𝛿 =
𝑚𝑔
𝑘
This represents the static deflection of the spring. Once the mass is displaced from the equilibrium
position and allowed to vibrate, the mass-spring system is no longer in equilibrium. Applying Newton’s
Second Law and simplifying:
𝛴𝐹 = 𝑚𝑎
𝑚𝑔 − 𝑘[𝛿 + 𝑦(𝑡)] = 𝑚𝑦̈ (𝑡)
𝑚𝑔 − 𝑘𝛿 − 𝑘𝑦(𝑡) = 𝑚𝑦̈ (𝑡)
𝑚𝑔
)−
𝑘
𝑚𝑔 − 𝑘 (
𝑘𝑦(𝑡) = 𝑚𝑦̈ (𝑡)
(8.1)
𝑚𝑦̈ (𝑡) + 𝑘𝑦(𝑡) = 0
(8.2)
Equation 8.1 is the governing equation of a frictionless spring-mass system.
The solution to this equation is:
𝑘
𝑚
𝑦𝑡𝑜𝑡𝑎𝑙 (𝑡) = 𝐴 cos (√ 𝑡)
𝑘
𝑚
The mass will oscillate as a cosine wave with amplitude A and angular frequency√ .
83
Laboratory 8 Differential Equations in Engineering: Spring-Mass Vibration
8.4 Procedure
Follow the steps outlined below after the Lab Teaching Assistant has explained how to use the
laboratory equipment.
1. Plug the axillary cable into a USB port on the computer not the monitor.
2. Load MATLAB, then make sure the springMass_myDaq.m script file has been downloaded from
the website and is in the current directory.
8.4.1 Displacement of Mass on One and Two Springs
1. Attach the spring to the stand and suspend the mass hanger from the other end. Add zero
weights (0 kg) to the mass hanger/cradle (0.033 kg).
2. Record the measurement from the scale. This is your reference value.
3. Complete Table 8.1 using one spring and two springs in series with six weights (0.321 kg). Wait
to calculate the spring constants k1 and k2.
Table 8.1: Displacement of Mass on One and Two Springs
One Spring
Mass with Cradle (kg)
0.033 (0 weights)
0.321 (6 weights)
Two Springs
yi (m)
Mass with Cradle (kg)
yi (m)
0.033 (0 weights)
0.321 (6 weights)
k1 =
k2 =
8.4.2 Oscillation - Maximum Weight (Six Masses and Cradle, 321 grams) and Two Springs
1. Place six (6) weights on the cradle and attach two springs to it. When stacking the weights,
ensure that each one is rotated 180 degrees from the weight above and/or below it. As a result,
the slots in each weight should not be facing the same side. This will help stabilize the cradle
and keep it centered in the shroud (i.e., the clear plastic cylinder/tube).
2. Ensure the cradle is centered in the shroud by looking down from the top of the tube. If
necessary, move the wooden stick to center the cradle.
3. Run the springMass_myDaq.m script file and wait for the prompt.
84
Laboratory 8 Differential Equations in Engineering: Spring-Mass Vibration
4. Use the string to pull the cradle down 0.5 to 1 inches. Simultaneously release the string and
press the ‘Enter’ button on the keyboard. NOTE: Avoid resting/pushing down on the table while
the program is collecting data.
5. Save the figure using a descriptive name.
8.4.3 Oscillation - Maximum Weight (Six Masses and Cradle, 321 grams) and One Spring
1. Use six (6) weights on the cradle and attach one spring to it. Repeat steps 1-5 from “8.4.2” for
the current setup.
8.4.4 Oscillation - Minimum Weight (Four Masses, 225 grams) and Two Springs
1. Use four (4) weights on the cradle and attach two springs to it. Repeat steps 1-5 from “8.4.2”
for the current setup.
8.4.5 Oscillation - Minimum Weight (Four Masses, 225 grams) and One Spring
1. Place four (4) weights on the cradle and attach one spring to it. Repeat steps 1-5 from “8.4.1”
for the current setup.
8.4.6 Data Analysis
1. Using measurements from Table 8.1, calculate the spring constants k1 and k2 with the
following equation. Attach hand calculations in Appendix.
𝑘𝑖 = |
𝑔(𝑚1 − 𝑚2 )
|
𝑦1 − 𝑦2
2. Analyze the figures from each of the above four oscillation cases. On each graph, ignore the
data from the first and last seconds of the run. Count the total number of peaks and
corresponding time (in seconds). Complete Table 8.2 on the next page.
a. For example, if you were to count seven (7) oscillations spanning 1.1 seconds to 6.5
seconds then you would produce a spring-mass system frequency of 7/5.4 or 1.3 Hz.
b. Use the frequency to determine the period for each case.
85
Laboratory 8 Differential Equations in Engineering: Spring-Mass Vibration
Table 8.2: Oscillation of Mass on One and Two Springs
One Spring
Two Springs
m = 0.225 kg (4 weights)
f (sec)
Tmeasured (sec)
f (sec)
One Spring
Two Springs
m = 0.321 kg (6 weights)
Tmeasured (sec)
f (sec)
Tmeasured (sec)
f (sec)
Tmeasured (sec)
3. The theoretical period Tcalc can be calculated by Equation 8.3. Find the calculated period for all
cases by completing Table 8.3.
𝑚
𝑇𝑐𝑎𝑙𝑐 = 2𝜋√ 𝑘
(8.3)
Table 8.3: Period of Each Oscillating Mass
k1
k2
m = 0.225 kg
Tcalc
Tcalc
k1
k2
m = 0.321 kg
Tcalc
Tcalc
86
Laboratory 8 Differential Equations in Engineering: Spring-Mass Vibration
8.5 General Guidelines (Executive Summary)
1. Write an Executive Summary for this lab using the Grading Guidelines on the following page
2. Answer the following questions within the body of the executive summary:
 Compare Tmeasured with Tcalc. Why are they different?
 Compare the stiffness of the one and two spring using the results of the lab.
3. Place Executive Summary, Appendix and Grading Guidelines in one word document and submit to
Carmen Dropbox. (10 Points will be deducted if the Grading Guidelines sheet is missing)
87
Laboratory 8 Differential Equations in Engineering: Spring-Mass Vibration
8.6 Integrals in Engineering Grading Guidelines (Executive Summary)
Section
Point
Breakdown
Header Information (See Tech. Comm. Guide)
Executive Summary
5
55
Clearly address the purpose of the examining the spring-mass system and
discuss the principles that were addressed throughout the lab.
12
Describe the main aspects of the lab procedure with proper transitions
between each step of the lab.
18
Overall, what did each of the tasks teach the team about second order
differential equations, spring constants, force, etc.
12
Incorporate answers to the lab questions within the body of the executive
summary. (See Section 8.5.2)
8
Clearly discuss the obstacles faced throughout the lab and address how
they were overcome. Did the team have problems with the equipment,
calculations, etc.?
3
Provide concrete examples of how to improve the lab procedure in order to
improve understanding of differential equations.
2
Writing Style
10
Grammar: See Technical Communication Guide for recommendations.
5
Organization and Progression: make sure the summary has natural breaks,
ideas are properly separated by paragraphs and topic sentences, and ideas
are fluently connected.
5
Appendix
30
Figures 1-4
12
Table 8.1, 8.2, 8.3
12
Hand Calculations for Table 8.1, 8.2, 8.3
3
Lab Participation Agreement
3
Group Grade
Points
Earned
100
88