Fall 2002 10 34 Numerical Methods Applied to Chemical Engineering Homework 4 Nonlinear algebraic equations and matrix eigenvalue problems Assigned Monday 10 6 02 Due Friday 10 11 02 Problem 1 Calculation of vibrational modes with eigenvalue analysis In this homework you will use a program that computes the normal vibrational modes of a 2D lattice see figure below using eigenvalue analysis A MATLAB program will be posted on the web site that calculates each vibrational mode and natural frequency for this system Another program will be provided that takes this information and makes a movie to animate a selected vibrational mode You will be asked to use these programs to compare the natures of the high and low energy vibrational modes of this structure No programming will be required FIGURE 1 2 D lattice of point mass spring system In this problem we wish to investigate the vibrational modes of an 2 D lattice of point masses connected by springs We assume that the lattice is infinite but we only want to simulate a small section of N rows and N columns of points shown in the figure above as the black filled circles To mimic the effect of having an infinitely large lattice of points we employ periodic boundary conditions in which we assume that the system is bounded by perfect copies of itself on all sides see figure below When one of the atoms in our system moves to the left its image also moves to the left In the figure below curved arrows link the atoms in the system filled circles to their images outside of October 7 2002 1 the system open circles For further details on how the periodic boundary condition is employed consult the attached MATLAB program FIGURE 2 Linking of atoms and their images in period boundary conditions to mimic an infinite lattice We now number the lattice sites from 1 to S N 2 as shown in the figure below FIGURE 3 Numbering system for lattice sites including identification of neighbor site bonding October 7 2002 2 While we will use the master indices for labelling purposes it is easier to write the potential energy function for the system if we label the site in column i and row j as i j The position of this site is x i j r i j EQ 1 y i j We write the total potential energy of the system as a sum of harmonic springs bonding neighboring lattice sites N 1 U r 1 1 r N N 2 N 2 2 1 i j i j K r i 1 j l b r i 1 j l b 2 EQ 2 i 1j 1 i j 2 i j 2 r i j 1 l b r i j 1 l b A factor of 1 2 is added before the summation to correct for the overcounting of each spring twice The distance between site i j and the image of site m n with which it interacts is i j i j 2 i j 2 1 2 r m n x m n y m n EQ 3 where we implement the periodic boundary conditions figure 2 as x i j x i 1 j i 2 N i j x i 1 j x 1 j x N j Nl b i 1 i j x i 1 j x i j x i 1 j i 1 N 1 x N j x 1 j Nl b i N EQ 4 y i j y i j 1 j 2 N i j y i j 1 y i 1 y i N Nl b j 1 i j y i j 1 y i j y i j 1 j 1 N 1 y i N y i 1 Nl b j N EQ 5 We subtract from each bond distance the preferred zero energy value of l b so that at mechanical equilibrium all lattice sites are separated by this distance The position in the minimum energy state of the i j lattice site that in column i and row j with master index k i 1 N j is October 7 2002 3 r i j x i j y i j i 1 l b EQ 6 j 1 l b We wish to compute the vibrational modes of the lattice and their natural frequencies To do so we pack all of the degrees of freedom of the system into a single vector of length F 2S q We also define a vector f of the forces on each degree of freedom that is the negative of the gradient of the potential energy q q1 x 1 1 q2 y 1 1 q3 x 2 1 f U U q1 U x 1 1 U q2 U y 1 1 U q3 U x 2 1 U y 2 1 U q4 qF 1 x N N qF y N N U qF 1 U x N N U qF U y N N q4 y 2 2 EQ 7 For the potential energy function above the derivatives of the potential energy with respect to the lattice site positions are computed in a straight forward if tedious manner 2 2 2 2 U K i j i j i j i j r i 1 j l b r i 1 j l b r i j 1 l b r i j 1 l b x i j 2 x i j i j i j r i 1 j r i 1 j i j i j U K 2 r i 1 j l b 2 r i 1 j l b x i j 2 x i j x i j i j 2 r i j 1 i j i j r i j 1 r i j 1 i j lb 2 r i j 1 l b x i j x i j i j i j i 1 j i 1 j EQ 8 x i 1 j x i 1 j i j U r ii j 1 j l b K r i 1 j l b i j i j x i j r r i j r i j 1 i j x i j 1 l b i j r i j 1 i j r i j 1 October 7 2002 i j x i j 1 l b i j r i j 1 4 i j i j i 1 j i 1 j y i 1 j y i 1 j i j i j U r i 1 j l b K r i 1 j l b i j i j y i j r r i j r i j 1 i j y i j 1 l b i j r i j 1 i j r i j 1 i j y i j 1 l b i j r i j 1 EQ 9 We define also a matrix of size F F of the second derivatives of …
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