Hi, g22-ga:
One term for such a rod, rigidly supported at one end only (the other
end being "free" but subject to load), is "cantilever beam". For a
general introduction to the fascinating topic of beams and their
deflections, see here, esp. the subsection entitled "Deflections":
[The Remarkable Theory of Beams, by J. B. Calvert]
http://www.du.edu/~jcalvert/tech/beam.htm
"Galileo studied beams, and although he did not get it quite right, he
showed how the subject should be approached. The theory of beams was
only perfected in the late 17th century with the rise of the science
of elasticity, and was shown to be a subject of great complexity for
which a full and accurate solution was very difficult."
A fully general treatment would need to manage not only the beam's
cross-section but also the possibility that although the rod itself is
of uniform composition, the material properties are anisotropic
(different in different directions). Some mention of these
difficulties is made here:
[Elasticity, by J. B. Calvert]
http://www.du.edu/~jcalvert/tech/elastic.htm
"Fortunately, most materials possess some symmetry in their
properties, and this symmetry can be used to reduce the number of
independent elastic constants."
A plane figure formed by a "uniform elastic rod" subject to varying
perpendicular load or "stress" along its (fixed) length is called an
elastic curve. The equation for an elastic curve has no general
solution in simple functions, although special cases (constant
curvature = circle) do. Historically the investigation of elastic
curves led to the topic of elliptic integrals:
[Physics to Mathematics: from Lintearia to Lemniscate]
http://www.ias.ac.in/resonance/Apr2004/pdf/Apr2004p21-29.pdf
[A Property of Euler's Elastic Curve]
http://math.tulane.edu/~vhm/papers_html/EU.pdf
However if as an approximation (accurate for small deflections) the
length of the curve is replaced by Cartesian coordinate x (taking y as
the deflection distance about the "neutral" x-axis), then a simplified
version is:
d^4 y / dx^4 = - w(x)/EI
where w(x) represents the load and EI is a constant. E (or sometimes
Y) is Young's modulus, and I is the moment of inertia about the
"neutral axis", here the x-axis; the product EI is known as the
"flexural rigidity".
If no weight is imputed to the rod itself, then w(x) would be zero
along the length, except at the very end of the rod. In this way one
can rationalize that the curve is a cubic polynomial:
y(x) = c_0 + c_1*x + c_2*x^2 + c_3*x^3
[If some modest uniform distribution of weight of the rod itself is
introduced, then in this simplified model an easily determined fourth
order coefficient is to be added.]
These coefficients can be determined from consideration of the
"boundary conditions" of the rod. At the rigidly supported end (x=0
using the most convenient coordinates) we fix both the location and
the slope of the rod, so that:
c_0 = c_1 = 0
At the other end of the rod we have a deflection -D (location) of the
rod's tip. However the slope is not specified there, so we must look
around for an additional boundary condition.
The simplest treatment would be to argue that if the rod were extended
beyond this point, then (again assuming weightlessness) its extension
would be straight. Continuity would then suggest the second
derivative is zero at the loaded/deflected end of the rod.
Let x = L be this end of the rod (recalling our approximation of the
length by Cartesian coordinate), so that our conditions are:
y(L) = -D
y"(L) = 0
and we have two equations to solve for c_2 and c_3:
c_2 + L * c_3 = -D/L^2
c_2 + 3L * c_3 = 0
One easily solves that:
c_3 = (1/2)D/L^3
c_2 = -3L * c_3 = -(3/2)D/L^2
* * * * * * * * * * * * * * * * * * * * *
The approximation of the shape of a flexible (but incompressible) rod
by a cubic polynomial, as we sketched above, is closely related to the
topic of splines and their convenient numerical proxies, cubic
splines. As in the discussion above, a true spline is parameterized
with respect to arc length while the cubic splines use a simpler
functional dependence on a Cartesian coordinate.
[The Curve of Least Energy]
http://people.csail.mit.edu/bkph/papers/Least_Energy.pdf
"In a thin beam, curvature at a point is proportional to the bending
moment. The total elastic energy stored in a thin beam is therefore
proportional to the integral of the square of the curvature. The shape
taken on by a thin beam is the one which minimizes its internal strain
energy. This is why we call the curve sought here the minimum energy
curve. A thin metal or wooden
strip used by a draftsman to smoothly connect a number of points is
called a spline."
In fact we can relate the "true spline" problem to that of the
cantilever beam in the following way. Where the cantilever beam is of
length L and passes from horizontal at the origin downward smoothly
through the "deflection" at some point (x,-D), consider the spline of
length 2L which passes through the symmetric pair of points (x,-D) and
(-x,-D). Such a spline is naturally symmetric with respect to the
y-axis.
Given a method for determining the true spline between these two
points of a given length L, we could vary the horizontal spacing (from
-x to x) until the resulting curve passes exactly through the origin.
The right-hand half of this curve would then be the "true" shape of
the cantilever beam.
regards, mathtalk-ga
* * * * * * * * * * * * * * * * * * * * *
Suggested Search Terms:
"cantilever beam"
"elastic curve"
"flexible rod"
"spline approximation" |
Request for Answer Clarification by
g22-ga
on
31 Jul 2005 11:00 PDT
Thank you mathtalk,
Your response is comprehensive and obviously adequate for the
question as asked. However, as this was the first question I have
asked on Google Answers, and not being cognizant of the complexity of
bending theory, I should have phrased my question more specifically.
You mention elastic curve, lintearia, and splines as possible curves
depending on material composition, etc. I found some confusion in
reading the attached articles as to where the distinction is made
between a deflection curve due solely to an applied force, as compared
to one due to the mass of the beam itself. The "elastic curve" seems
to refer to the latter.
I am in need of a designation of the curve generated by deflected rod
due only to a perpendicularly applied force, as would be exhibited by
a deflected floor or ceiling mounted cantilever. I would now ask:
What kind of curve is generated by a deflected aluminum cantilever
(strongly isotropic material)? Assume a very small value of
deflection, Less than 1% of the length of the cantilever, and assume
no other forces (gravity or mass of the bar itself) contribute to the
bending. A simple answer as to the most appropriate name of curve
should suffice. With the name of the curve, I can then explore the
articles again, as it relates to the specific cantilever I just
outlined. Thank you.
|
Clarification of Answer by
mathtalk-ga
on
31 Jul 2005 14:11 PDT
Hi, g22-ga:
The phrase "elastic curve" is an inclusive one and certainly applies
to cases in which all the bending is attributed to discrete external
loading (rather than the "body forces" acting along the length of the
curve due the beam's own weight).
However as a way of emphasizing the lack of such influences, I would
suggest the combined phrase "thin-beam elastic curve".
The phrase "cantilever beam" would denote the arrangement you describe
of being rigidly mounted (not "hinged") on the side of a building. If
the rod does not extend horizontally, you might clarify this point by
elaborating this as an "inclined cantilever beam".
In the "alternate" terminology of splines (which are "thin" by
definition), the usual prescription is just the length of the spline
and the location of two endpoints (or possibly additional points
through which it is made to pass). If you wish to use this kind of
language to describe your curve, it would be proper to say "a
(two-point) spline, clamped at one end". This implies that the slope
as well as the location is being prescribed, but only at one (clamped
spline would tend to be read as meaning clamped at both ends, which is
not your case).
regards, mathtalk-ga
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