Transcript Magnus Effect

Student Introduction:
Name: Blake Loeb
School: Eau Gallie High School
Grade: 11th
FLVS: Hope
M. Blake Loeb is enrolled in the Cambridge Program and has a strong interest in
Mathematics, Physics, Computer Simulations and Aerodynamics.
In this year project he was able to combine these interest to create a Magnus Effect
Simulator which utilized Visual Basic and Excel to perform Numerical Integration (Runge
Kutta 4th Order) to create a predictive model of the Magnus effect on 6mm spherical
projectiles.
The study of the trajectories of spherical projectiles is a key area of
experimentation for classical physics as described by Sir Isaac Newton’s
laws of motions.
These laws however did not take into account the impact of the forces
of aerodynamics (Lift & Drag). Heinrich Gustav Magnus (1802 – 1870),
who investigated the effect experimentally in 1853 on rotating cylinders.
Source: http://pencilcricket.blogspot.com/
As a result the lift generated by a spinning spherical projectile is now
called the Magnus effect. The Magnus effect is still being research today
as there are many complex factors that influence the Magnus effect.
In order to generate backspin to produce the Magnus effect spacers can
be used to transfer angular velocity to the spherical projectile.
The goal of this experiment is to test different geometries of spacers
and create a mathematically model of the lift generated by the Magnus
effect from the experimental results.
To achieve a successful model of the Magnus force this project will
cover the mathematics of applied physics for ballistics spanning
hundreds of years.
Photo taken by student researcher
Hypothesis: A concave based spacer will be the most
consistent way of applying pressure to a spherical
projectile to cause backspin and generate optimal lift
Question:
a)
Which hop up spacer geometry will create the greatest range and lift from the Magnus effect ?
a)
What set of equations, variables. algorithms and processes can be used to create an accurate
predictive model to generate the resulting spherical projectile trajectory?
Included
in Testing
Included
in Testing

Included
in Testing
Used to create backspin
Forces Governing Trajectory
1. Force Equations
2. Key Variables
3. Gravity
4. Velocity
5. Drag Force Equation
6. Lift Equation (Magnus Force)
7. Drag & Lift Coefficient
8. Launch Angle & Velocity
9. System of Differential
Equations

What vector forces govern the trajectory of a
spherical projectile? (including drag & lift)
where F is the resulting force, Fg is the force from gravity, Fd is the
force from drag, Fl is the force of lift from the Magnus effect and
Fc is the force from all the rest such as additional interactions due
to angular momentum.
For simplicity this analysis will focus on the dimensions of
distance and height from these forces and will not consider cross
winds or additional interacts (Fc )
Source: Physics of Paintball: Dr. Gary Dyrkacz
Source: Flight mechanics of a Spinning Spheroid :
John C. Adams, Jr., Ph.D

How long will it take a spherical
projectile to fall 3 feet? (Time in
flight without lift or air
resistance)
X = gt2/2 where
x=distance, g=32.174
feet/sec2 and t=time
Solving for time where
distance = 3 feet
t=Square root
(3*2/32.174)
Time = .432 seconds
What will be the velocity be at a
given point in time ?
vf = vi + a * t
where Vf is the final velocity,
Vi, is the initial velocity
a is the acceleration/deacceleration
with t as the time in seconds
Velocity changes in a non-linear
fashion with an exponential decay
Source: “Ideal Lift of a spinning sphere."
Aerodynamics of a Sphere. Benson, Tom NASA
- Glen Research Center, Jul 28 2008

What are the equations
governing the Drag Force?
Fd is the force from drag, Cd is the
experimentally determined coefficient
of drag, ρ is the density of air, A is the
two dimensional area of the object,
which for a sphere is the same as the
area of circle, π d2/4 and v is velocity
of the projectile .

What are the equations
governing the Lift Force?
Fl is the force from lift, Cl is the
experimentally determined coefficient
of lift, ρ is the density of air, A is the
two dimensional area of the object,
which for a sphere is the same as the
area of circle, π d2/4 and v is velocity
of the projectile .
Source: Physics of Paintball: Dr. Gary Dyrkacz



The Magnus effect is a function of RSR
P is pressure, R is the gas constant, T is absolute
Temperature, and rho is air density of the air
CL and CD are determine experimentally
Source: Flight mechanics of a Spinning Spheroid :
John C. Adams, Jr., Ph.D

1.
Drag impact to V:
dV/dt = -Fd g /m – g sin α
2.
Lift impact to L
dL/dt =( FL g /m – g cos α)/V
3.
Launch Angle
These coupled differential equations are solved by numeric
integration (Runge Kutta) in the Magnus Effect Simulator
Source: Flight mechanics of a Spinning Spheroid :
John C. Adams, Jr., Ph.D
Procedure: Target placement will be used to generate a projectile trajectory. Multiple tests will be executed with
calculation of consistency, averages, standard deviation, and grouped by spacer geometry type..
Procedure Steps:
1.
A mathematical model will be developed using vector and kinetic equations to predict X & Y groupings of a
spherical projectile.
2.
Air will be used to propel small spherical projectiles (6mm) down a chamber with an exchangeable spacer to
produce backspin for .2g , .28g and .4g BBs
3.
A chronograph will be used to measure exit velocity. Adjustment will be made so that for each test to make
sure the propulsion system is level and targeted. This will mean that changes in X & Y groupings will be from
the Magnus effect from the backspin produced.
4.
Targets that have pressure sensitive indicators will be used to determine the X & Y groupings at regular
intervals (50 feet).
5.
The X & Y grouping information will be plugged into the mathematical model to determine lift and relative rate
of backspin.
6.
By each spacer geometry type the results will be ranked by the lift generated. Graphical and statistics based
analysis will be applied. The Hypothesis that the concave based spacer will generate the most lift and
consistent grouping will be tested and determine if it is correct. These results will be discussed and a
conclusion will be created from this data.
Constants
Independent
Dependent
Gravity Force
Spacer Type
Maximum Range*
Gas Constant of Air
Spacer Depth
Maximum Height*
Gas Constant Conversion Factor
Time
Height at time t
Launch Angle (0 degrees)
BB Weight
Distance at time t
Launch Height
Initial Velocity (fps)
BB Diameter
Angular Velocity (rpm)
BB Surface Area
Rotational Spin Ratio
Pi
Rho
Pressure
Reverse Magnus Region %
Temperature
Lift Coefficient
Drag Coefficient
* Response Variable

Type of flow is dependent on the Rotational Spin Ratio
(Angular Velocity (Spin)/Forward Velocity and Sphere
Surface Smoothness
Source: J. Hoffman and C. Johnson, Computational
Turbulent Incompressible Flow, Springer 2007.
Reverse
Magnus
Effect:
Laminar flow
Negative
Lift
Magnus Effect: Positive Lift
Turbulent flow
Source: Airsoft Trajectory Project

Reverse Magnus experimentally determine to
occur between 100% - 80% of initial velocity for
6 mm diameter BBs weighing .2 - .4 g


Actual versus Simulation Example
Created using Excel & Visual Basic
Height in Inches
Trajectory .2 g BB by Geometry Type
100.00
50.00
Concave .2
Flat .2
0.00
-50.00
0
50
100
150
200
250
300
350
H Shape .2
Range in Feet
Height in Inches
Trajectory .28 g BB by Geometry Type
100.00
Concave .28
50.00
Flat .28
0.00
-50.00
0
50
100
150
200
250
300
350
H Shape .28
Range in Feet
Height in Inches
Trajectory .4 g BB by Geometry Type
150.00
100.00
Concave .4
50.00
Flat .4
0.00
0
50
100
150
200
Range in Feet
250
300
350
H Shape .4
MAXIMUM
VALUE
Range in
Feet
Concave .2 g
Height in
inches
Concave .28 g
Height in
inches
Concave .4 g
Height in
inches
100
43.25
42.00
28.20
150
53.75 Japa
53.83
49.67
66.00
65.00
65.30
250
68.75
69.00
90.00
300
52.50
50.75
93.50
325
12.50
1.00
87.00
350
0.00
0.00
25.50
375
0.00
0.00
1.00
200
Concave Spaced Geometry achieved the most range (375 Feet) and the most
height (93.5 inches).
Concave Spaced Geometry also had the closest grouping with the minimum
standard deviation.
1. The hypothesis is correct and the concave spacer
was the most consistent way of applying pressure
to a spherical projectile to cause backspin and
generate optimal lift (maximum range & height)
2. The goal of this experiment to create a
mathematically model of the lift generated by the
Magnus effect from the experimental results was
achieved with the Magnus Effect Simulator.
Books
Internet
Butterworth, Heinemann, Aerodynamics for Engineering Students,
Fifth Edition 2008 by E. L. Houghton, P. W. Carpenter.
(Paperback 9780750651110)
"Ideal Lift of a spinning sphere." Aerodynamics of a Sphere. Benson, Tom
NASA - Glen Research Center, Jul 28 2008. Web. 28 Sep 2010.
J. Hoffman and C. Johnson, Computational Turbulent
Incompressible Flow, Springer 2007
Robins, B., Mathematical Tracts, 1 & 2. J. Nourse, London, 1761.
Robins, B., Investigation of The Difference in The Resisting Power
of The Air to Swift and Slow Motion, 1742
Journals and Magazines
Robins-Magnus Effect: A Continuing Saga.,Tapan K. Sengupta &
Srikanth B. Talla Dept. of Aerospace Engineering
L.J. Briggs, "Effect of spin and Speed on the Lateral Deflection
(Curve) of a ball; and the Magnus Effect for Smooth Spheres,"
American Journal of Physics V 27, pp. 589-596 (1959). Journal of
Fluid Mechanics
The Magnus or Robins effect on rotating spheres Journal of Fluid
Mechanics (1971), 47: 437-447 Cambridge University Press
Newton, I. (1672), New theory of light and colours. Philosophical
Transactions of the Royal Society London, 1, 678-688.
Acknowledgments
My thanks to the staff of SOCOM Airsoft Arena, 210 NW 13th
Street, Ocala, FL 34475, for allowing me to perform this testing at
their facility.
http://www.grc.nasa.gov/WWW/K-12/airplane/beach.html
" Physics of Paint Ball" Lennon, Tom. Fresno University, Jul 28 August 24
2000
http://lennon.csufresno.edu/~nas31/nsa/pballIntro1.html
" The Magnus Effect Equation" , Tom Cull, Staff, Clinical Sciences MR
Division, Picker International Science, Jul 28 1999
http://www.madsci.org/posts/archives/1999-06/928944018.Ph.r.html>
" Maximizing the Range with Newton's Method" James E. White, Calculus in
Action Webbook
Project Welcome to Calculus, Jul y 2004
www.maa.org/projectwelcome/calculus%20in%20action.html
" Magnus Effect Experiment" , Julian Rubin, Encyclopedia of Aviation,
December 2009
http://www.juliantrubin.com/encyclopedia/aviation/magnus_effect.html
“Effect of Hop-Up and the Magnus Effect”
The Airsoft Trajectory Project, cybersloth.org , December 2009
http://mackila.com/airsoft/ATP/03-a-01.htm
“Forces Governing Trajectory” The Airsoft Trajectory Project, cybersloth.org
, December 2009
http://www.physics.armstrong.edu/faculty/mullenax/research/riseball.html
“Magnus effect on Spherical Projectiles” By Sanjay Mittal and Bhaskar Kumar
Department of Aerospace Engineering, Indian Institute of Technology, Kanpur,
UP 208 016, India, J. Fluid Mech. (2003), vol. 476, pp. 303{334. 2003
Cambridge University.