Dynamics

Transcription

Dynamics

www.EngageEngineering.org
Everyday Examples in Engineering
Dynamics
This material is based upon work supported by the National Science Foundation (NSF) under Grant
No. 083306. Any opinions, findings, and conclusions or recommendations expressed in this
material are those of the author(s) and do not necessarily reflect the views of NSF.
Table of Contents:
Quarter-Mile Time of a Dragster
3
Constant Acceleration: Mini Cooper Drag Race
15
Projectile Motion: X-15 Aloft
17
Friction & Newton’s Laws: Superman 11th Hour
19
The Wet-Dog Shake: Overcoming Surface Tension with Centripetal Force
20
Freefall: Falling Hare
23
Engineering Elements Embedded in a Child’s Toy
24
Snakes on A Plane
32
Figure Skater
35
Wanted: Everyday Examples in Engineering
37
2
ENGAGE is an Extension Services project funded by the National Science Foundation. The overarching
goal of ENGAGE is to increase the capacity of engineering schools to retain undergraduate students by
facilitating the implementation of three research-based strategies to improve student day-to-day classroom
and educational experience. ENGAGE strategies include:
• Integrating into coursework, everyday examples in engineering (E 3s)
• Improving student spatial visualization skills
• Improving and increasing faculty-student interaction
For more information about ENGAGE contact:
Susan Metz, Project Director
[email protected]
For more information on E3s or to submit E3s
contact:
Pat Campbell, Ph.D.
[email protected]
Jennifer Weisman, Ph.D.
[email protected]
Go to EngageEngineering.org for lesson plans for the following courses: Calculus and Differential
Equations, Chemistry, Circuits, Control Systems, Dynamics, Elasticity and Plasticity, Engineering Design,
Engineering Graphics, Fluids, Introduction to Engineering, Manufacturing, Material Failure, Mechanics,
Physics, Properties of Materials, Statics, Stress and Strain and Thermodynamics.
3
Using Everyday Examples in Engineering (E3)
A Dynamics Project: Quarter-Mile Time of a Dragster
Shannon Sweeney
Penn State Erie, The Behrend College
This is an applied assignment for undergraduate students in a dynamics course. The
assignment relates to an interest of many engineering and technology students: drag
racing. Students are given enough information about a popular sports car model, and
some simplifying assumptions, to estimate its time in the quarter-mile. Students can then
compare their results to published drag racing results for that vehicle. Even with the
simplifying assumptions, the calculated results are strikingly close to the published
results.
INTRODUCTION
The purpose of this project was to make a comprehensive assignment that is real and
interesting to engineering and technology students. Many assignments in a dynamics
course are strictly academic in nature, not attracting the students’ interest and often
leaving the students wondering about the application. Most engineering and technology
students have at least a cursory interest in drag racing and some may be full-fledged
enthusiasts or participants. Consequently, this assignment should have broad appeal in a
dynamics course.
Students are given measured engine performance data for a Chevrolet Corvette C605. They are given the approximate values for the weight of the car, its transmission shift
schedule, its differential ratio, its tire diameter, and a common method for estimating both
rolling resistance and wind resistance. Students are also given the actual drag racing
results published for a 1999 Chevrolet Corvette Coupe. It is not known if the weight,
transmission, differential, and tire data given to the students matches that of the car for
which race data is published but they are assumed to be close enough for the purpose of
this assignment.
The assignment is comprehensive in that it requires utilizing the relationship of power to
force and acceleration in free-body and kinetic diagrams, requires modeling of dynamic
resistive forces, and requires finding displacement and time associated with non-constant
acceleration. The five race data that are published include 1/4 Mile ET (elapsed time),
1/4 Mile MPH (miles per hour), 1/8 Mile ET, 1/8 Mile MPH, and 0-60 Foot ET. By
carefully establishing a spreadsheet to carry out the appropriate integrations, students
should able to closely match all five race data.
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THE MODEL
Free-Body and Kinetic Diagrams
By modeling the car as a particle and by equating free-body and kinetic diagrams,
students should be able to show:
a = (1/m)(P/v-FR-FW)
where a = instantaneous acceleration of the car,
m = mass of the car (students are given a weight W of 3200 lb),
P = instantaneous power (from engine performance data),
v = instantaneous velocity of the car,
FR = rolling resistance (later assumed constant), and
FW = wind resistance (later assumed proportional to square of velocity).
Engine Performance Data
Students are given the following tabulation of engine speed and wheel horsepower which
is available from RRI [1]. Full-throttle operation is assumed for the curve.
Engine Speed Wheel Power
n (rpm)
P (hp)
1751
100.6
1993
118.3
2493
151.3
2994
186.7
3496
219.9
3999
267.8
4214
281.3
4401
295.9
4603
309.0
5006
331.3
5507
346.0
5810
353.9
6010
353.8
6212
350.2
6429
346.6
5
The tests conducted by RRI determine wheel horsepower which is the power available at
the wheel to propel the vehicle forward. By using wheel horsepower, drivetrain
inefficiencies have been accounted for.
Transmission Shift Schedule
Students are given the following transmission shift-up schedule:
Gear Ratio Start rpm Shift rpm
1
3.27
1700
5500
2
2.20
3700
5500
3
1.56
3900
5500
4
1.22
4300
5500
5
1.00
4500
5500
6
0.82
4500
~
The ratio is transmission input (engine output) speed to transmission output speed.
Differential Ratio and Tire Diameter
Students are given a differential ratio of 3.08. This ratio is differential input
(transmission output) speed to differential output (wheel input) speed. Students are given
a tire diameter of 25.66 inches.
Rolling Resistance
Rolling resistance is simply modeled with a coefficient of rolling friction fr [2]. A value
of 0.012 is given for fr which is a commonly assumed value for pneumatic rubber tires on
the hard surface of a drag strip. FR is then equal to frW and students should be able to
show that it is constant at 38.4 lb.
Wind Resistance
Wind resistance is modeled as a quadratic loss or a velocity-squared loss [3]. Students
are given the following relationship and values:
FW = (CρA/2)v2
where FW = wind resistance,
C = drag coefficient (≈ 0.35 for sleek sports cars),
ρ = fluid (air) density (accepted as 0.0807 lbm/ft3 @ STP),
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A = frontal area (≈ 24 ft2; based on the approximate width and height), and
v = velocity of the car.
The biggest challenge for students will be proper handling of units. For the given
information, students should be able to show:
FW = 0.01053v2
where FW = wind resistance force in lb,
v = instantaneous velocity of car in ft/s,
and the units for the constant, 0.01053, are lb-s2/ft2.
Simplifying Assumptions
Some simplifying assumptions stated for the students are:
1) Engine performance data, although produced during steady-state conditions,
applies to acceleration conditions.
2) Automatic transmission and neglect shift times.
3) Torque converter locked up at all times (no torque multiplication).
4) Neglect mass moments of inertia of wheels, drivetrain, and engine.
5) Sufficient tire grip coefficient to prevent slip.
6) Neglect tire growth.
7) Assume a velocity of 12.9 mph at start (engine at 1700 rpm with transmission in
first gear). The actual practice of starting a race usually involves running the
engine to about 4000 rpm with the brakes locked and the torque converter stalled,
and then releasing the brakes. Students at the freshman or sophomore level will
not have the background to model the behavior of the torque converter and its
effect on acceleration. Therefore the assumption of a 12.9 mph head start
approximates the actual practice when starting a race.
The simplifying assumptions mean that engine speed n and car velocity v will be directly
proportional, with the proportionality depending on the transmission gear.
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THE ANALYSIS
With the given data and simplifying assumptions, students should be able to show:
a = 0.01006(P/v – 38.4 – 0.01053v2)
where a = instantaneous acceleration of car in ft/s2,
P = instantaneous wheel power in ft-lb/s,
v = instantaneous velocity of car in ft/s,
and the units for the constant, 0.01006, are lb-s2/ft or slugs.
Wheel power P will be a function of car velocity related through the engine data,
transmission gear, differential ratio and tire diameter. Consequently, car acceleration a
will be non-constant and it will be a determinable function of car velocity v.
Initially, it may appear to be advantageous to make car velocity v the controlled variable
throughout the project and find car acceleration a as a function of car velocity. However,
transmission shifts are based on engine speed, not car velocity. As a result, the author
has found it advantageous to make engine speed the controlled variable throughout the
project and find car acceleration a and car velocity v as a function of engine speed
n. Either approach must utilize the assumption that engine speed and car velocity are
directly proportional.
With engine speed being the controlled variable throughout the project, students will
need wheel power expressed as a function of engine speed. This can be accomplished
several ways but students are encouraged to develop an equation for wheel power as a
function of engine speed from the provided tabulated engine data. Students should find
that a third-order polynomial as follows fits quite well as shown in Figure 1.
P = -4.04e-9n3 + 4.08e-5n2 – 0.0567n +99.2
where P = wheel horsepower, and
n = engine rpm (revolutions per minute).
This equation has a coefficient of determination r2 of 0.9993, meaning that it can be used
to accurately determine wheel power from engine speed in the range: 1700 rpm < n <
6500 rpm.
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Figure 1. Engine Performance Data
With engine speed as the controlled variable, students will need to express the car
velocity as a function of engine speed. The function is determined by the transmission
gear, differential ratio, and tire diameter. Figure 2 shows the relationships students will
need to develop and utilize to express v as a function of n.
n/v
dv
(engine rpm/
(ft/s)
Gear
car ft/s)
(dn = 200 rpm)
1
89.96
2.223
2
60.52
3.305
3
42.92
4.660
4
33.56
5.959
5
27.51
7.270
6
22.56
8.865
Figure 2. Relationship of Engine Speed to Car Velocity
With all things considered, the instantaneous acceleration throughout the race is a
determinable function of instantaneous velocity. So, for any engine speed and each
transmission gear, the instantaneous car velocity and the instantaneous car acceleration
can be determined.
Since instantaneous acceleration is a function of instantaneous velocity, the following
relationships will be necessary:
a = dv/dt, leading to an integration of
dt = (1/a)dv
and
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a = vdv/ds, leading to an integration of ds = (v/a)dv.
Continuous functions for (1/a) and (v/a) can be derived from the curve-fit equation of the
engine data and the information in Figure 2 but the functions are rather complicated. The
functions could be integrated by closed form techniques but are best handled by a
tabulation and subsequent use of the trapezoidal rule.
By using engine speed as the controlled variable throughout the project, students will
need to develop a tabulated integration for time t as the engine goes from start rpm to
shift rpm for each transmission gear. Students will also need to develop a tabulated
integration for displacement s as the engine goes from start rpm to shift rpm for each
transmission gear. From the resulting tabulation of t and s (each as a function of engine
speed and transmission gear), the five race data can be determined.
SAMPLE ASSIGNMENT
Although the assignment can be given several ways, the author has chosen to give all the
necessary information to the students at the beginning of the project and allow students
about four weeks to complete the project. The information provided is the vehicle
weight, engine performance tabulation, transmission schedule, differential ratio, tire
diameter, coefficient of rolling friction, and wind resistance information which includes
the drag coefficient, frontal area, air density, and the quadratic loss equation. Simplifying
assumptions are also given and discussed.
To assist the students, the author has broken the project into six parts and uses a small
amount of class time discussing each part, about every other class period. The first
category is development of appropriate free body and kinetic diagrams.
The second part is development of the horsepower equation as a function of engine
speed. Curve-fitting procedures and options within spreadsheets are discussed with the
students. Students are asked to plot the data and their curve-fit as shown in Figure
1. They should find a maximum horsepower of about 350 and that it occurs at about
6000 rpm. From this, students need to realize that the stated horsepower of an engine
occurs at a specific speed and that the engine does not have the stated horsepower at all
speeds.
Students should establish a spreadsheet using engine speed as the controlled variable,
showing start rpm to shift rpm for each transmission gear. Engine speed increments of
200 rpm are encouraged. Students may be provided a spreadsheet template with this
much information so they start the project correctly and uniformly. Students must then
find the wheel horsepower for each engine speed from the engine performance
equation. It should be converted to appropriate units of ft-lb/s and students are
encouraged to create spreadsheet columns for power in both units.
10
The third part is development of the vehicle velocity in ft/s as a function of engine speed
as shown in Figure 2. The roles of the transmission, differential, and tire are discussed
with the students. The car velocity should be converted to familiar units of mph and
students are encouraged to create spreadsheet columns for velocity in both
units. Throughout this part, students should be able to consider the transmission schedule
and show that the transmission output speed at shift rpm for any gear is the same as it is
for the start rpm at the next higher gear.
The fourth part is modeling the force from wheel horsepower and the dynamic resistive
forces for the free body diagram. The relationship of power and velocity and the subjects
of rolling friction and quadratic loss wind resistance are reviewed with the students. The
instantaneous values of each force can be determined as a function of engine speed.
The fifth part is development of acceleration. The instantaneous values of rolling
resistance and wind resistance are used in conjunction with the wheel force and mass m
(=W/g) to get the instantaneous acceleration in ft/s2 as a function of engine
speed. Acceleration should be converted to familiar units of g’s and students are
encouraged to create spreadsheet columns for acceleration in both units. Also, the
necessary integration relationships are discussed, revealing the quantities 1/a and v/a as
those to be integrated.
The sixth part is the integration of 1/a and v/a respectively. Students can determine the
instantaneous values of 1/a and integrate that quantity with respect to velocity, by
utilizing the trapezoidal rule, to find values of elapsed time t. Students can also
determine the instantaneous values of v/a and integrate that quantity with respect to
velocity, by utilizing the trapezoidal rule, to find values of distance traveled s. For each
integration, and using 200 engine rpm increments, students will have to develop and
utilize the values of the differential term dv shown in Figure 2. From the tabulation of t
and s, the five race data can be determined. The completed spreadsheet is shown in
Figure 3.
Given the amount of assistance provided in the six parts, students are made aware of the
published results shown in Figure 4 upon completion of the project. With the published
results, students can assess their performance in the exercise.
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12
Figure 3. Completed Spreadsheet with Integration
13
CONCLUSION
The appropriate race data can be interpolated from the tabulation of t vs. s and compared
to published [4] data for a similar car. Figure 4 summarizes the results.
Figure 4. Comparative Race Data
From the tabulation of calculated results in Figure 3, some meaningful, realistic
observations can be made. The car will reach 60 feet of travel shortly after it shifts into
second gear. The car will reach an eighth of a mile, or 660 feet, while it is in fourth
gear. The car will reach a quarter mile, or 1320 feet, shortly after it shifts into fifth gear.
The acceleration of the car is non-constant but varies only slightly from engine start rpm
to engine shift rpm within each gear. The acceleration decreases with increasing
gear. These are common observations for full-throttle acceleration in any car.
The car acceleration in first gear is about 1 g. Depending on the weight distribution and
the center-of-gravity height, the front wheels will have little or no contact with the road
during first gear acceleration which is a common observation for many
dragsters. Consequently, most of the vehicle weight will be on the rear wheels during
first gear acceleration and the tire grip coefficient will need to be about 1.0 to prevent
slip, Figure 5.
14
Figure 5. Corvette Acceleration: Courtesy of Edgar Perez (car driver/owner) and Michael Yoskich
(photographer)
The wind resistance becomes significant, especially at speeds greater than 100
mph. Even for a sleek Corvette, the wind resistance force at 100 mph is over 220 lb.
The top speed is estimated at about 175 mph which is realistic for this Corvette. At this
speed, the summation of the resistance forces from rolling and wind equal or exceed the
force available from the engine to propel the vehicle forward. Acceleration goes to zero
because force equilibrium has been reached.
This assignment should be of great interest to most engineering and technology
students. It should provide an exciting and familiar application of many dynamics
principles. Consequently, students should be motivated to spend enough time on the
project to develop a better understanding of various dynamics principles.
REFERENCES
1. Rototest Research Institute, Salemsvagen 20, S-1444 40 RONNINGE, SWEDEN
www.rototestinstitute.org
2. Tongue, B.H. and S.D. Sheppard (2005). Dynamics – Analysis and Design of
Systems in Motion. New York, NY: John Wiley & Sons
3. Inman, D.J. (2002). Engineering Vibration. Upper Saddle River, NJ: Prentice-Hall
4. DragTimes.com, P.O. Box 827051, Pembroke Pines, FL 33082-7051
www.dragtimes.com
© 2010 Shannon Sweeney. All rights reserved. Copies may be downloaded from
www.EngageEngineering.org. This material may be reproduced for educational
purposes.
15
This material is based upon work supported by the National Science Foundation
(NSF) under Grant No. 083306. Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the author(s) and do not
necessarily reflect the views of NSF.
.
Constant Acceleration: Mini Cooper Drag Race
Chad Young
Nicholls State University
Link to video clip: http://www.archive.org/details/MiniCooperDragRace (you can mute
sound if you wish)
.
16
This clip shows races between Cooper Mini’s, one with a standard 5-speed transmission
and the other with an automatic 6-speed transmission. The distance for these trials is ¼
mile, which is approximately 400 m. The times are given on the upper section of the
electronic billboard, which the cameraman conveniently focuses on.
What is the acceleration of the cars?
First, have your students write down all that they know:
The times of the 2 cars are t1 = 16.42 s and t1 = 15.07 s.



The distance of travel is 400 m.
Assume constant acceleration throughout the race.
The initial velocity and position are zero.
Now, the appropriate kinematics equation is:
Solving for the acceleration, we find the acceleration for the two cars
You can also solve for their final velocities:
Substituting the accelerations and times for the two cars:
Since the mini cooper certainly does not have uniform acceleration over the course of the
track, have the students ask and answer where the acceleration is likely greatest (at the
beginning of the course) and if it is greater or less than the acceleration they calculated
(greater) and why.
17
© 2010 Chad Young. All rights reserved. Copies may be downloaded from
purposes.
18
Projectile Motion: X-15 Aloft
Chad Young
Link to video clip: http://www.archive.org/details/X-15Aloft
The X-15 was a research aircraft created by the agency that would eventually become
NASA. It set a number of records for speed and altitude of flight. In this clip, the X-15 is
released from a B-52 “mother plane” and falls back to Earth.
Before showing the clip, ask your students, if a package is dropped from a plane, which
of the following paths will it take if air resistance is negligible:
Of course, the correct answer to this is D. The package maintains the same x-component
of velocity as it had in the plane. This is demonstrated as well in the X-15 clip. The X-15
remains under the B-52 as they both travel forward at the same speed.
In addition, ask your student how the dropped package would appear to someone inside
the plane? (The package will appear to drop straight down.)
U.S. Air Force videos of bomb drops can also illustrate these concepts:
http://www.youtube.com/watch?v=bYXdk-qTl5U and
19
http://www.youtube.com/watch?v=fLTEIfzVrEg Both show the bombs keeping more or
less level with the plane until something happens.
Discussion questions and activities for these videos include:
• Why are the unpowered bombs keeping pace with the plane?
• Why are they moving down?
• When do they start moving backwards?
• Are they moving backwards?
• Draw force diagrams for the bombs to explain their motion.
purposes.
20
Friction & Newton’s Laws: Superman 11th Hour1
Chad Young
Link to video clip: http://www.archive.org/details/Superman11thHour
In this 1942 episode of Superman, the hero sabotages the Yokohama Navy Yard in Japan.
Superman, as a character, was generally not involved in acts of war, but this WWII era
cartoon obviously acted as propaganda against U.S. enemies.
This classic cartoon also offers a nice point of discussion about Newton’s Laws of Motion1
and the forces of friction. Superman uses his great strength to force the ship into the bay
where it sinks. To do so, he must overcome the frictional force between the ship and rails as
well as the force required to break the ropes. While Superman might be strong enough to do
such things, the frictional force between his feet and the ground, which acts as the opposite
and equal force to that exerted by Superman, is clearly not sufficient to hold him in place.
Which force is greater: the frictional force between the ship and rails or between Superman
and the rails? Clearly, the normal force acting on the ship is much greater than that acting on
Superman, so, unless Superman has super sticky feet, his frictional force is much less than
that acting on the ship.
www.EngageEngineering.org. This material may be reproduced for educational purposes.
This material is based upon work supported by the National Science Foundation (NSF) under
1material are those of the author(s) and do not necessarily reflect the views of NSF.
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21
The Wet-Dog Shake: Overcoming Surface Tension with Centripetal Force
Andrew Dickerson, David Hu
Georgia Institute of Technology
Link to video of High Speed Shaking: http://www.youtube.com/watch?v=kvoKN1UfLn0
Imagine exiting a shower into a cold environment with no option to towel dry. A human
would face hypothermic danger in this scenario. Without the ability to towel dry or reach
most parts of their bodies, furry mammals, from mice to grizzly bears, have evolved the
ability to shake. The high-speed shaking oscillations are initiated at the head and travel
down the loose skin toward the tail. The accelerations generated by the animal on its fur
acts to shed water from wet fur, and is so effective, mammals can remove over half the
water entrained in the fur within a few seconds.
Think of an animal’s body cavity as cylinder with radius R. Let us establish a polar
coordinate system in the cross-section of a dog’s body where an angle of rotation θ, is
measured normal to the ground.
The centripetal force an object on the cylinder’s (dog’s) surface is given by Equation 1
Fcent  mR2rˆ
where m is the object’s mass (water drops in our case), and  is the spinning frequency
measured in rad/s. Through observation, it has been seen that a point on the center an
animal’s back, between it’s shoulders will rotate through   / 2 rad during oscillatory
shaking. An average sized dog has a body cavity radius, measured just aft of the front
legs, of 12 cm and shakes at about 4.5 Hz. This equates to twisting its skin through 28.3
rad/s. In order to generate a similar centripetal force, smaller animals must shake faster.
A mouse, with body radius of 1.25 cm, shake at 30 Hz! The biggest mammals, such as
bears, shake around 4 Hz. Since animals cannot quickly change their radius, genetics and
muscular dynamics choose a sufficient frequency based on size. There seems to be little
dependence of shaking frequency on species.
22
(1)
Evaluate: calculate the acceleration ( acent  R2 ) of the average dog. Convert the units
to m/s2 and calculate the number of g’s the skin, and water trapped inside, experiences.
Let us think of centripetal force as an outward force for now. What is a mammal trying to
overcome by shaking so violently? The surface tension of water causes it to coalesce with
itself, adhere to many surfaces, and resist external forces. Surface tension is the culprit
for water drops sticking to your window and wetting your clothes. Water finds the
position of lowest energy in every situation.
Centripetal force is actually a force that keeps an object in its path of rotation. This may
be counter-intuitive, but centripetal translates to “center-seeking.”
Experience: stand on the center of a carousel and have a friend spin it at a near constant
rotational velocity. Increase your radius to the center by moving outwards. What do you
feel?
The force that our hands and feet provide, keeps us planted on a spinning carousel. We
experience what feels like a force pulling us outwards that is actually the tendency of our
mass to continue in straight-line motion. If you were on the edge of a carousel of radius r
and spinning with angular velcoity and let go, you would eject in a straight line at a
velocity given by Equation 2
(2)
The water ejected from a dog’s fur also ejects in a straight line.
Picture a particle on a dog’s back as it shakes. The path the particle travels can be
modeled as a sine wave
23
  A sin t
where A is the maximum angle of deflection ( ~  / 2 rad). The second derivative of
Equation 3 will allow for the calculation of the maximum acceleration the skin
experiences. The looseness of the skin allows it to deflect much greater angles than the
body rotates, which is about  / 6 rad.
Elaborate: the interested student will calculate the number of g’s the skin experience on
direction change for a given A where sin t 1.
Engineering Application: Think about how we typically dry clothes in a washing
machine. Could this be made better by what we’ve seen in nature?
For more information visit:
dickerson.gatech.edu
All images and graphics by: Andrew Dickerson, Georgia Institute of Technology
© 2010 Andrew Dickerson and Chad Hu. All rights reserved. Copies may be downloaded
from www.EngageEngineering.org. This material may be reproduced for educational
purposes.
24
(3)
Freefall: Falling Hare
Chad Young
Link to video clip: http://www.archive.org/details/FallingHare
This 1943 Merrie Melodies cartoon, with Bugs Bunny, has a scene where Bugs and a gremlin
are in an airplane that is headed straight towards the ground. The plane is in freefall for over 1
minute. You can have your students calculate the distance the plane covers in that time
(disregarding air resistance):
and the plane’s final velocity:
which is approximately 1300 mph despite what the airspeed gauge says.
In a more general sense, use this clip to introduce freefall motion. Ask your students some of
these questions:
•
•


•


What happens to the speed of the plane as it drops?
What is the effect of air resistance?
Was there evidence of air resistance in the cartoon?
If the force of air resistance were equal to the force of gravity, what would happen?
What causes the acceleration? If the plane’s engines stop, will the plane stop?
What would happen to the plane if the force due to gravity went away? Would the
plane stop (since there is no force), or would it keep going?
What would happen if the force of air resistance were greater than the force of
gravity?
purposes.
This material is based upon work supported by the National Science Foundation (NSF) under
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under Grant
No. is
083306.
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findings,
conclusions
or recommendations
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083306.
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opinions,
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recommendations
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NSF.
25
Introduction to Civil Engineering: Engineering Elements Embedded in a Child’s
Toy
Indranil Goswami
Morgan State University
Photo used with permission of International Playthings LLC
Objective of the game “Frog Hoppers”
The game Frog Hoppers is a simple child’s (3+) toy whose objective is to use elastic
spring-like element (the frog’s tail) to make the plastic frog jump into a plastic bucket,
which also serves as the container. The frogs come in 4 different colors, so that 4
individuals or teams can make it a competitive game.
The objective of this exercise is to discover elements of engineering in this toy. Believe it
or not, there are about 4 to 5 groups of engineering concepts embedded in the design of
this simple toy.
We are going to review the following concepts through our learning modules:
1. Measurements – use of precise equipment (Vernier caliper) for measurement
2. Kinematics – projectile motion
3. Structural Mechanics – bending stiffness of beams
4. Work and Energy – conversion of elastic (spring) energy into kinetic energy
5. Dynamic Coupling – between support and structure
6. Strength of Materials – fatigue strength
7. Aerodynamics – drag resistance
26
Learning Module 1: Measurements
Use a Vernier caliper to measure the following dimensions. See figure 4.1 to identify
these variables.
1. Length of the frog tail, L (mm)
2. Width of the frog tail, b (mm)
3. Thickness of the frog tail, t (mm)
4. Mass of the frog, m (mg)
LEARNING OBJECTIVE 1
List the measured variables in table 1 below:
Table 1: Summary of physical properties
Variable Name Value
Units
L
b
t
m
Learning Module 2: Kinematics
Once a ‘particle’ is released in the gravitational field with a velocity Vo as shown in
figure 2.1 below, the theoretical path of travel is parabolic. If the origin is located at the
point of launch, the equation of the parabola is

g sec 2  
y  x  tan  
x
2Vo2


Equation 2.1
Vo
H

R
Figure 2.1: Projectile Motion of a Particle in a Gravitational Field
Since the launch angle  and launch velocity Vo are initial values (constant), this
equation describes a parabola.
27
The theoretical range (R) of the projectile can be obtained by setting y = 0 in the equation
R
Vo2 sin 2
g
Equation 2.2
The theoretical maximum height (H) of the projectile can be obtained by setting dy/dx =
0 in the equation
H
Vo2 sin 2 
2g
Equation 2.3
Since it is difficult to measure the launch velocity Vo, use the ratio of equations 2.3 and
2.2 as shown below to estimate the launch angle 
H Vo2 sin 2 

R
2g
Vo2 sin 2 1
 tan 
g
4
Equation 2.4
Measure H and R for each launch and estimate launch angle  using equation 2.4
Since the measurement of R is more reliable than H, use equation 2.2 to calculate the
launch velocity Vo using the measured R and the estimated 
For launches 1 and 2, push the tail-spring all the way down to the floor. Since this gap
can be pre-measured, this gap is equal to Δ. For launches 3 and 4, place a dime on the
floor and push the tail-spring to the top of the dime. Since the thickness of the dime can
be pre-measured, the difference between the original gap and the thickness of the dime is
equal to Δ. For launch 5, use a quarter in a similar way. Summarize your results in table 2
below.
Table 2: Summary of kinematic data for launches 1-5
Launch
H
R
Δ
 4H 
  tan 1 

No.
(mm)
(mm) (mm)
 R 
1
2
3
4
5
28
Vo  gR sin 2
Learning Module 3: Structural Mechanics – Bending Stiffness of Beams
Figure 3.1 shows a cantilever beam (fixed at one end and free at the other end) loaded
with a transverse point load at the free end. This is how we are modeling the frog’s tail.
Also, see figure 4.1
P

Figure 3.1
Bending of a Cantilever Beam under Tip Load
Elastic theory can be used to demonstrate that the (maximum) tip deflection of a
cantilever beam is given by

PL3 4 PL3

3EI Ebt 3
Equation 3.1
where E is the modulus of elasticity of the beam material. Typically, the modulus of
elasticity or Young’s modulus is determined from a tensile test. The initial slope of the
stress-strain diagram from such a test is the modulus of elasticity E.
A steel (E = 200GPa) ruler which has a thickness of 1 mm and width of 25 mm is rigidly
clamped at one end and pushed with a steady, static force by placing a mass of 1 kg on
the end in a manner similar to the figure above. Calculate the static deflection of the end
(mm).
Calculate the stiffness of the beam described above. Note that the stiffness of the beam is
a property of its material and section geometry, and is independent of the load.
Learning Module 4: Application of Work and Energy Principles
The frog is launched into motion by pushing down on its tail, thereby storing elastic
energy. Upon release, this energy is converted into kinetic energy, resulting in the frog
attaining a launch velocity Vo. If it is assumed that the potential energy of the frog at the
instant of launch is zero (neglecting the slight depression below the datum), then equating
the spring elastic energy to the frog’s kinetic energy, we have
mVo2
1 2 1
2
k  mVo  k  2
2
2

Equation 4.1
29
The spring stiffness k is the elastic stiffness of a cantilever beam given by
k
3EI 3E bt 3 Ebt 3
 3

L3
L 12
4 L3
Equation 4.2
L
b

Figure 4.1
k
t
Dimensions of Various Elements of the Elastic Spring (Tail)
4mVo2 L3  4mL3  Vo2 
Ebt 3 mVo2


E

  3  2 
4 L3
2
bt 3 2
 bt   
Equation 4.3
The mass of the frog (m) is to be measured precisely, Vo has been previously estimated,
L, b and t are the length, width and thickness of the tail spring – all of which can be
measured quite precisely using the Vernier caliper.
Thus, using the measured parameters m, L, b, t and Δ and the calculated parameter Vo,
we can estimate the elastic modulus E for the plastic material.
For each of the launches tabulated in module 2, the vertical deflection of the tail-spring is
controlled (see next page) and recorded in the last column of table 3.
For each launch, the modulus of elasticity is calculated using the following equation
 4mL3  V 2 
E   3  o2 
 bt   
Equation 4.4
Rewrite the data from table 1 into the first 5 columns of table 3 below. For launches 1
and 2, push the tail-spring all the way down to the floor. Since this gap can be premeasured, this gap is equal to Δ. For launches 3 and 4, place a dime on the floor and push
the tail-spring to the top of the dime. Since the thickness of the dime can be premeasured, the difference between the original gap and the thickness of the dime is equal
to Δ. For launch 5, use a quarter in a similar way.
30
Table 3
Launch H
No.
(mm)
R
(mm)

(degrees)
Vo
(mm/sec)
Δ
(mm)
E
(GPa)
Learning Module 5: Dynamic coupling between support and structure
The lid of the plastic bucket doubles as a launching surface. The added elasticity of the
lid serves to increase the range (R) of the frog. By using two separate launch positions for
the frog, make multiple measurements of the range.
Procedure:
1. Mark two positions on the launch lid – one exactly at the center of the lid and
another from a rigid surface such as a tabletop
2. Make 5 launches from each position. In each case measure the range and compute
the average range. Also compute the standard deviation of the measurements and
comment on the variation of these measurements.
3. Compare the range from the flexible surface to the range from the rigid surface.
Comment.
Learning Module 6: Strength of Materials – Fatigue Strength
You must appreciate that in the choice of material for the frog, material behavior under
repeated loading must play an important role. How a material responds to the repeated
cyclic loading and unloading as opposed to supporting a sustained load for a long time is
key in determining the durability of the material and the long-term quality of the toy.
Research online using the keywords ‘fatigue strength’ and write a definition for it.
Learning Module 7: Aerodynamics – Drag Resistance
The kinematic equations presented in section 2 and then used in this paper are based on
particle kinematics, i.e. for an object that can be modeled as a particle (occupying no
space). For such an object, there would be no aerodynamic drag. However, our plastic
frog does have significant drag potential and therefore, particle kinematics is not strictly
valid. As a result, our conclusions do have a certain degree of error.
In designing experiments, it is always important to understand the various sources of
systematic error that are embedded in the methodology (in addition to random human
error that can always occur, even in perfectly designed experiments). The presence of
such errors, produced as a result of simplifying assumptions, does not completely
31
invalidate the experiment, as long as the results are significant and these errors are within
some acceptable bounds.
© 2010 Indranil Goswami. All rights reserved. Copies may be downloaded from
purposes.
This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 083306. Any
opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not
32
Snakes on a Plane
Hamidreza Marvi and Dr. David Hu
Georgia Institute of Technology
Engage: If you happen to have access to a (non-poisonous) snake you can bring the
snake to class. Inviting Samuel L. Jackson to bring a snake to class would be even better
but probably not a realistic option. Since this really won’t be possible for most people, a
fake snake is a great substitute or a video of a snake trying to move on a slippery surface:
http://www.youtube.com/watch?v=WquF95sEbCk
Bring a fake snake (if you don’t want use a snake, pretty much any object, even a book
could potentially work), something to serve as a plane (preferably made of a rough
material), and a ruler to the class.
Explore: Ask students about the role of friction in their life. Ask them if they can walk
on ice. Why not? Ask what happens if you put a heavy book and a light book on a plane
and tilt the plane until they slide down the hill. Which one do they think slides first? Ask
them if they have any ideas about how to measure the friction coefficients of an object
sliding on another object.
Explain: Friction plays a major role in locomotion of all animals, including snakes.
Snakes cannot move on slippery surfaces. To investigate this hypothesis we put a snake
on a very slippery surface. As shown in the following video (the same one from the start
of class), the snake is not able to move forward:
http://www.youtube.com/watch?v=WquF95sEbCk
Snakes also need different friction coefficients in different directions to be able to move.
To investigate this hypothesis, we put a jacket on snake that has similar frictional
properties in all directions. The snake cannot move at all as shown in the following video:
http://www.youtube.com/watch?v=B2Dswa8f3T4
Now the question is, how snakes can modify their frictional properties in different
directions? The following videos show the role of their scales on friction modifications:
http://www.youtube.com/watch?v=e1_Sk1pC1gY
http://www.youtube.com/watch?v=kw8vRuncg2c
33
Figure 1 shows the snake scales while snake is being pulled towards its head (a) or tail
(b). The snake scales are assembled in such a way that they can grip the substrate in one
direction (Figure 1b).
Figure 1. Scales of a corn snake
Elaborate:
In order to elaborate on the role of friction on the locomotion of snakes we need to be
able to measure their friction coefficients on a substrate. We need to measure friction
coefficients in both forward (head down the hill) and backward (tail down the hill)
directions to test our hypothesis. We can use an inclined plane experiment to do this
measurement as shown in Figure 2. The object should be placed on the plane and then we
should tilt the plane until the object starts sliding down the hill. The height corresponding
to that specific angle needs to be measured in order to find the friction coefficient. This
will be explained in more details in the following.
Figure 2. Inclined plane experiment
34
Before we write the force balance on snake, we should remember that the forces applied
to any object on a plane are mgsin( ) towards down the hill and mgcos() normal to the
plane, where m is mass of object, g is gravitational acceleration and  is the inclination
angle. We should also know that the friction force is defined as F  N where  is the
friction coefficient and N is the normal force applied to the object. Now, we can write
 the snake on the plane in the longitudinal

the force balance for
direction as following:


mg sin(  )  m
g cos(  )




(1)
If we simplify this equation we can find the following equation for friction coefficient:
(2)
 tan()
H
Now the only parameter that needs to be found is  . We know that sin( )  . Thus 
L
can be found using the following equation:
H
  sin 1 ( )
(3)
L


Therefore, in order to find the friction coefficient of any object sliding on a plane we
need to put the object on that plane and tilt the plane until the object slides down the hill.
We should measure the height that the object starts sliding down as well as the length of
the plane. Then using equation 3 we can calculate the inclination angle. We can then use
equation 2 to find the friction coefficient of the object sliding on the plane.
We used the same simple experiment to measure the friction coefficients of snakes in
different directions. The friction coefficient of a corn snake on Styrofoam in the
backward direction is almost twice as that of forward direction.
Evaluate:
Example 1: What happens if you put a heavy book and a light book on the plane and tilt
the plane until they slide down the hill. Which one slides first?
The friction coefficient is not related to the mass. What matters are the surface properties
of the object and the plane.
Example 2: Put an object on the plane provided and measure its friction coefficient using
the equations 1 and 3.
All images and graphics by: Hamidreza Marvi, Georgia Institute of Technology
© 2011 Hamidreza Marvi and David Hu. All rights reserved. Copies may be downloaded
from www.EngageEngineering.org. This material may be reproduced for educational
purposes.
35
This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 083306. Any
opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not
36
Angular Momentum, Moment of Inertia: Figure Skater
Chad Young
© 2010 Chad Young. All rights reserved. Copies may be downloaded from www.EngageEngineering.org. This material may be
reproduced for educational purposes.
Link to video clip: http://www.archive.org/details/FigureSkaterAngularMomentum
This segment from a news program details the training regiment for Dana Carellas, a 15year old figure skater. In a number of scenes, she performs a spin; as she comes out of the
spin, she extends her arms out to the side to slow down. This is a typical problem
utilizing the conservation of angular momentum in which the object’s moment of inertia
changes so the angular velocity changes as well.
First, consider how her moment of inertia changes. Initially, Dana can be best represented
by a spinning cylinder. Assuming her mass to be 50 kg and her “radius” to be 0.2 m, her
moment of inertia is:
Ask students to make models to determine what shapes represent Dana with arms in and
arms out (when Dana extends her arms out, her body is best approximated by a solid
cylinder rotating about its central axis - her body- and a thin rod about an axis through
center perpendicular to the length). Assuming her armspan (from fingertip to fingertip) to
be 1.5 m and the mass of her arms to be 8 kg, her new moment of inertia is:
Now, consider that Dana is spinning at 2 revolutions per second when her arms are pulled
in. What is her new angular speed when she extends her arms?
37
The law of conservation of angular momentum states:
So, Dana can decrease her angular speed by a factor of 2. What can she do to increase her
angular speed? What are some other things she can do with her body to alter her speed?
It is also helpful to have a turntable in the classroom to allow students to check their
calculations through experiment, as seen in this video:
http://www.youtube.com/watch?v=yAWLLo5cyfE.
38
Wanted: Everyday Examples in Engineering (E3s)

Who: ENGAGE (Engaging Students in Engineering), funded by NSF, is
developing a library of Everyday Examples in Engineering (E3s). E3s are
needed for 1st and 2nd year engineering courses, especially Introduction to
Engineering, Circuits, Engineering Graphics, and Fluids.

What: E3s are objects that are familiar to students that can be used to
demonstrate/teach engineering concepts. Examples of E3s include using hula
hoops and wooden rulers to teach free or forced vibrations; sausages
to demonstrate Mohr's circle of stress; or iPods that demonstrate many
things. E3s aren’t design challenges or projects. Examples of E3s can be
viewed at www.EngageEngineering.org in the resources section.

Why: Integrating E3s into 1st and 2nd year engineering courses is one of the
research-based strategies ENGAGE is using to improve retention of
engineering students. Studies indicate that students are motivated to learn
when they can make a connection to something familiar to them which
illustrates the concept they are trying to understand.

How: The initial submission does not need to be in any particular format. It
can be a description of the Everyday Example and the specific concept that it
demonstrates. More extensive write-ups are also welcome. E3s will be
reviewed and if accepted, project staff will work with the author on revisions
so there is sufficient detail to be used effectively and easily by faculty.

Honorarium & Recognition: The author(s) of each accepted E3 will receive
a $150 thank you honorarium. In addition, a letter will be sent to the
author’s department chair and dean. The letter will include reference to the
author’s contribution to ENGAGE by having their submission selected under a
peer review process, and made available for national and international
dissemination. Authors will maintain their E3 copyright and give permission
to ENGAGE to use the materials.

Where: For more information or to submit E3s, contact Dr. Patricia Campbell
([email protected], 978-448-5402) or Dr. Jennifer Weisman
([email protected]).
39

Dynamics

Transcription

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