Download Using Matlab to Calculate Top Performance

Battling
Top
Design:
http://www.youtube.com/watch?v=-Uw5crBtIqs
http://www.youtube.com/watch?v=AvEwpLV8-R4&NR=1
What makes a good Intro Project
competitive outcome
wide design space
low tinkering coefficient
low math ability
use of SolidWorks/Matlab/Excel
low material cost
low fabrication skill
link to science related topics
safety
Coil Gun Ranking
competitive outcome
wide design space
low tinkering coefficient
low math ability
use of SolidWorks/Matlab/Excel
low material cost
low fabrication skill
link to science related topics
safety
SCORE
2
2
2
10
5
2
9
9
5
46
"Battling Top" Ranking
competitive outcome
wide design space
low tinkering coefficient
low math ability
use of SolidWorks/Matlab/Excel
low material cost
low fabrication skill
link to science related topics
safety
SCORE
7
7
7
10
8
8
9
9
10
74
Catapult Ranking
competitive outcome
wide design space
low tinkering coefficient
low math ability
use of SolidWorks/Matlab/Excel
low material cost
low fabrication skill
link to science related topics
safety
SCORE
9
7
7
10
6
6
7
9
5
66
Rules
• Core Material limit
– 2"x2"x2" aluminum cube
– plus anything you want to screw/fasten into it
• like a shaft for launching
• Design own launcher (acrylic possible)
• Rip cord < 24"
• Arena will be some bowl shaped thing
– anyone have a snow disk?
•
http://www.youtube.com/watch?v=uerlueDxUqM&feature=related
Tops involve angular rotation, a
challenging engineering topic
• Moment of inertia
• From Wikipedia, the free encyclopedia
• In classical mechanics, moment of inertia, also called mass
moment of inertia, rotational inertia, polar moment of
inertia of mass, or the angular mass, (SI units kg·m²) is a
measure of an object's resistance to changes to its rotation.
It is the inertia of a rotating body with respect to its
rotation. The moment of inertia plays much the same role
in rotational dynamics as mass does in linear dynamics,
describing the relationship between angular momentum
and angular velocity, torque and angular acceleration, and
several other quantities. The symbol I and sometimes J are
usually used to refer to the moment of inertia or polar
moment of inertia.
Moment of Inertia
• Overview
• The moment of inertia of an object about a given axis describes
how difficult it is to change its angular motion about that axis.
Therefore, it encompasses not just how much mass the object has
overall, but how far each bit of mass is from the axis. The farther
out the object's mass is, the more rotational inertia the object has,
and the more force is required to change its rotation rate. For
example, consider two hoops, A and B, made of the same material
and of equal mass. Hoop A is larger in diameter but thinner than B.
It requires more effort to accelerate hoop A (change its angular
velocity) because its mass is distributed farther from its axis of
rotation: mass that is farther out from that axis must, for a given
angular velocity, move more quickly than mass closer in. So in this
case, hoop A has a larger moment of inertia than hoop B.
• The moment of inertia of an object can
change if its shape changes. Figure skaters
who begin a spin with arms outstretched
provide a striking example. By pulling in their
arms, they reduce their moment of inertia,
causing them to spin faster (by the
conservation of angular momentum).
• Scalar moment of inertia
• Consider a rigid body rotating with angular velocity ω around a
certain axis. The body consists of N point masses mi whose distances
to the rotation axis are denoted ri. Each point mass will have the
speed vi = ωri, so that the total kinetic energy T of the body can be
calculated as
• In this expression the quantity in parentheses is called the moment
of inertia of the body (with respect to the specified axis of rotation).
It is a purely geometric characteristic of the object, as it depends
only on its shape and the position of the rotation axis. The moment
of inertia is usually denoted with the capital letter I:
• It is worth emphasizing that ri here is the distance from a point
towards the axis of rotation, not towards the origin. As such, the
moment of inertia will be different when considering rotations
about different axes.
• Similarly, the moment of inertia of a
continuous solid body rotating about a known
axis can be calculated by replacing the
summation with the integral:
• where r is the radius vector of a point within
the body, ρ(r) is the mass density at point r,
and d(r) is the distance from point r to the axis
of rotation. The integration goes over the
volume V of the body.
Ycm, height of center of mass
• Ycm is the average height of mass distribution
making up the top. if you wanted to balance it
from a string, sideways, Ycm is where you
would attach the string:
Ycm
Stability Coefficient
• The overall stability of the top, based on some
approximations, is indicated by:
• stability =
Rg/Ycm
– Radius of Gyration divided by
– the height of the center of mass
• Implications:
– increase Rg, stability ++
– decrease Ycm, stability ++
The calculations are daunting
• If we wanted to try a number of different
designs, it would take forever
• Fortunately engineers use computer tools
such as Matlab to speed the development of
design