This exercise considers the problem of finding the traverse displacement ( )

Order Details:
This exercise considers the problem of finding the traverse
displacement () of an elastic beam, which is
subject to a body force ().
The potential energy is
               
where is a positive constant. It is assumed
that the beam has what are called simply supported ends. One consequence of
this is that (0) = 0 and (1)
= 0. The role of the boundary conditions will be discussed in more detail in
part (f).
(a) Suppose the grid points are 0
= 0, 1 = , 2
= 2, ··· , n+1
= 1, where  =
»
This exercise considers the problem of finding the traverse
displacement () of an elastic beam, which is
subject to a body force ().
The potential energy is
               
where is a positive constant. It is assumed
that the beam has what are called simply supported ends. One consequence of
this is that (0) = 0 and (1)
= 0. The role of the boundary conditions will be discussed in more detail in
part (f).
(a) Suppose the grid points are 0
= 0, 1 = , 2
= 2, ··· , n+1
= 1, where  = 1/(+
1). Writing  write down the composite trapezoidal
approximation for  .
(b) Use a centered second-order approximation for xx
at 1,2,
··· , n. You can use a
first-order approximations for xx at 0
and n+1. Using these
with the result from part (a), what is the resulting approximation for  ? In
doing this, remember that 0 = n+1
= 0.
(c) What is the resulting matrix equation that must be
solved to find the minimum for the approximation for  found
in part (b)?
(d) The minimum of  from part (b) can be found using the
MATLAB command where  is the approximation from part (b)
and U is an -vector containing a starting guess
for (1, 2,
··· , n)T
. What would be a good, simple, and nonzero choice for U, and why is it
a good choice? In answering this, it is worth knowing that a cable hanging
between two poles is an elastic beam, subject to gravity.
(e) If  = 1 and  =
4, then the exact solution
is
               
On the same axis, plot the exact solution and the numerical
solution when using 22 grid points. In doing this, explain how you found the
numerical solution (using part (c) or part (d)), and why.
(f) A simply supported beam is required to satisfy the four
boundary conditions: (0) = 0, (1)
= 0, xx(0) = 0, and xx(1)
= 0. The numerical solution was derived without any mention of the last two
conditions. The reason is that they are natural boundary conditions, which
means that if  minimizes  , then it will
automatically satisfy xx(0) = 0
and xx(1) = 0. In
comparison, (0) = 0 and (1)
= 0 are essential boundary conditions, which means that we must explicitly
require this of our approximation (see part (b)). So, the question is, does
your numerical solution satisfy (approximately) xx(0)
= 0 and xx(1) = 0. Using
the first-order approximations you used in part (b), from the numerical
solution calculate xx(0) and xx(1),
using  = 20, 40, 80, 160. Are your answers
consistent with the statement that the minimizer satisfies xx(0)
= 0 and xx(1) = 0?
»

• WE OFFER THE BEST CUSTOM PAPER WRITING SERVICES. WE HAVE DONE THIS QUESTION BEFORE, WE CAN ALSO DO IT FOR YOU.
• Assignment status: Already Solved By Our Experts
• (USA, AUS, UK & CA PhD. Writers)
• CLICK HERE TO GET A PROFESSIONAL WRITER TO WORK ON THIS PAPER AND OTHER SIMILAR PAPERS, GET A NON PLAGIARIZED PAPER FROM OUR EXPERTS

QUALITY: 100% ORIGINAL PAPER – NO PLAGIARISM – CUSTOM PAPER

Why Choose Us?

How It Works

• Click on the “Place Your Order” tab at the top menu or “Order Now” icon at the bottom and a new page will appear with an order form to be filled.
• Fill in your paper’s requirements in the “PAPER DETAILS” section.
• Fill in your paper’s academic level, deadline, and the required number of pages from the drop-down menus.
• Click “CREATE ACCOUNT & SIGN IN” to enter your registration details and get an account with us for record-keeping and then, click on “PROCEED TO CHECKOUT” at the bottom of the page.
• From there, the payment sections will show, follow the guided payment process and your order will be available for our writing team to work on it.

About AcademicWritersBay.com

AcademicWritersBay.com is an easy-to-use and reliable service that is ready to assist you with your papers 24/7/ 365days a year. 99% of our customers are happy with their papers. Our team is efficient and will always tackle your essay needs comprehensively assuring you of excellent results. Feel free to ask them anything concerning your essay demands or Order.

AcademicWritersBay.com is a private company that offers academic support and assistance to students at all levels. Our mission is to provide proficient and high quality academic services to our highly esteemed clients. AcademicWritersBay.com is equipped with competent and proficient writers to tackle all types of your academic needs, and provide you with excellent results. Most of our writers are holders of master’s degrees or PhDs, which is an surety of excellent results to our clients. We provide assistance to students all over the world.
We provide high quality term papers, research papers, essays, proposals, theses and many others. At AcademicWritersBay.com, you can be sure of excellent grades in your assignments and final exams.