EGM5315 Fall 2018

Department of Mechanical and Materials Engineering

This is Dr. Levy’s EGM5315 Intermediate Analysis of Mechanical Systems Fall 2018 page

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Here is the (8/01/18) updated syllabus for the course.

My office is in EC3442, and email address is levyez@fiu.edu

My tel. no. is 305-348-3643. My fax no. is the department fax no. 305-348-1932

Office hours: TBD and announced at the end of the first week of classes

TA: none.

Photocopies of the book for this class will be available in my office starting Aug. 20. Please come to my office to get the materials. Cost is $45 per set. Please use your own copy of any ODE book and read up on first and second order systems. I will give you handouts from other ODE books to supplement the lectures.

Here is the video of the first lecture

Here is the video of the second lecture This deals with solutions of 2^nd order ODEs

This material and all the linked materials provided, except where stated specifically, are copyrighted © Cesar Levy 2018 and are provided to the students of this course only. Use by any other individual without written consent of the author is forbidden.

Here is the video of the third lecture This deals with series solutions

Here is the video of the fourth lecture This continues series solutions of 2^nd order ODEs

HW problems found in the handouts were discussed on Wednesday Aug 29 to clarify what was required to obtain a solution and all problems discussed were announced as being due Wednesday Sept 5. Please submit your work neatly.

Here is the video of the fifth lecture This deals with solutions near the point at infinity (up to 23 minutes in the tape). Afterwards, we begin discussing PDEs. We derive the heat equation for a 3-D non-isotropic body, then using constraints we reduce the equation for an isotropic body and for the case of 2-D and 1-D bodies, including the steady state case.

Here is the video of the sixth lecture which discusses the method of characteristics for first order systems, does an example, and also discusses the method for second order systems.

Here is the video of the seventh lecture which looks at solution of second order equations, canonical forms and reduced canonical forms.

Here is the video of the eighth lecture discussed classical wave, heat and Laplace Equation in the context of canonical forms, boundary and initial data needed, well- and ill-posedness, 1^st, 2^nd, 3^rd type of BCs.

Don’t worry about the name of the lecture; these lectures were taped a while back but cover the materials we are covering now in class

Here is the video of the ninth lecture discussed well- and ill-posedness, 1^st, 2^nd, 3^rd type of BCs. Also discussed in this video is the topic of Separation of Variables (SOV) as a means of solution which will be touched upon.

We plan to complete the solution of the wave equation and begin discussing the wave equation in polar coordinates which is covered in the video of the tenth lecture. We plan to discuss solution of the 2-D wave equation in polar coordinates using SOV. We will discuss Bessel’s equation and solutions, BCs for same.

Here is the video of the twelfth lecture discussed solution of the 1-D wave equation in Cartesian coordinates with initial conditions. We discussed use of Fourier series as a means of solution for the unknown coefficients.

Here is the video of the thirteenth lecture discussed solution the 2-D wave equation in polar coordinates using SOV with initial conditions. We discussed use of Sturm-Liouville System as a means of solution for the unknown coefficients.

We announce the first examination for October 17. This exam will cover solution of ordinary differential equations using series solutions. Characterization of partial differential equations (elliptic, parabolic and hyperbolic), determining characteristics, reducing PDEs to canonical forms and also to reduced canonical forms. You will have access to your notes.

Lesson 15 – Here is the video of the fifteenth lecture the Sturm-Liouville solutions in more detail. We showed how to find the denominators of the unknown coefficients. Showed how this led to the denominators of the Bessel equation coefficients, the denominators of the Fourier coefficients. We started solution of inhomogeneous PDE with inhomogeneous BCs.

Here is the video of the sixteenth lecture. We obtained solution of inhomogeneous PDE with inhomogeneous BCs by the method of splitting. Found steady state solution that takes care of inhomogeneous part of the PDE and the inhomogeneous part of the BCs. Found transient solution.

We are having the first exam 10/17 covering Picard, Series solutions, Method of Frobenius, method of variation of parameters, method of undetermined coefficients, characterization of singularity points, method of characteristics, canonical forms, reduced canonical forms, and from the separation of variables constructing eigenfunctions and finding eigenvalues.

Here is the video of the seventeenth lecture. We reviewed the splitting problem. We showed how to get the initial condition for the transient solution and how to evaluate the unknown coefficient of the transient solution, by finding both the top and bottom integrals.

Wednesday 10/18: your first exam was a take home exam and covers materials up to and including reduced canonical equations, and determining eigenfunctions and eigenvalues. The exam will be due on 10/22 at noon.

Here is the video of the eighteenth lecture; We cover how one can get solution for the case of inhomogeneous BCs by splitting and solving a simple problem that takes care of the BCs and a second problem whose equation is now inhomogeneous but whose BCs are homogeneous.

Here is the video of the nineteenth lecture; we begin discussion of Laplace transforms and its use; what are the requirements on the function to be transformed, discussed some properties of Laplace transforms; solved an ODE problem and started a PDE problem to show how the transform is modified to solve PDE problems. Please start reading about use of Laplace transforms in Chapter 10.

Here is the video of the twentieth lecture. We finished the problems that we started last time. We also did a problem with an integral boundary condition. A portion of the video of the lecture is lost because of a power outage but the sound is still there. The video comes back at the end of the tape.

Here is the video of the twenty-first lecture. We solved the wave equation problem using Laplace Transforms. We discussed the Residue Theorem and complex integration to obtain the result.

Here is the video of the twenty-second lecture. We discuss the sine, cosine and Fourier transforms and the changes to watch out for using these transforms. Please read Chapter 7 on Fourier Transforms.

Here is the video of the twenty-third lecture. We discussed the sine, cosine and Fourier transforms and solved two problems in class, one using sine transform and one using cosine transforms.

Second exam is announced for November 21. It will cover separation of variables methods with homogeneous BCs, non-homogeneous BCs, non-homogeneous PDE, Sturm-Liouville and Orthogonality, Laplace transforms, Fourier sine, cosine, and Fourier transforms.

Here is the video of the twenty-fourth lecture. We will begin discussing Fourier Transform as a manner of solving the biharmonic equation Ñ⁴f=0 in a semi-infinite region. Please start viewing this tape from the 27^th minute until the end.

Here is the video of the rest of the twenty-fourth / and beginning of twenty-fifth lecture. We continue discussing use of the Fourier Transform. We solve the half-space problem with a delta function load in the y direction and no load in the x.

In this videotape we also suggest getting the other stresses for this problem and solving the delta function load in the x direction and no load in the y direction.

Nov 21 exam 2

We continued discussing use of the Fourier Transform. We solve the half-space problem with a delta function load in the +y direction and no load in the x on the positive have space y>0. We also solved for the stresses s_xy, s_yy, and showed you how to get the stresses s_xx. We showed you how to find the stresses for the second situation where the d(x) function was taken parallel to the x axis. We also made mention of how to solve the general situation when s_yy=f(x) as a convolution using the solution for the delta function as the kernel for the green’s function.

Here is the video of the twenty-seventh lecture. We start discussing self-similar solutions.

Here is the video of the twenty-eighth lecture. We discuss a second self-similar problem with a non-constant boundary condition.

Final exam will be given on Nov 28 as a take home exam. It will cover all the materials given in the videos, so please view the videos.