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Homework 3: Vibrations of Atoms in a Solid

1. Problem Statement & Equations of Motion

A simple model for illustration of the vibration of atoms in a solid is the number \(n\) of identical masses (\(m\)) connected by springs (all with the same spring constant \(k\)) along a line.

The equations of motion are:

$$ m \frac{d^2u_1}{dt^2} = k(u_2 - u_1) + C e^{i\omega t} $$ $$ m \frac{d^2u_i}{dt^2} = k(u_{i+1} - u_i) + k(u_{i-1} - u_i), \quad 2 \le i \le n-1 $$ $$ m \frac{d^2u_n}{dt^2} = k(u_{n-1} - u_n) $$

Substituting \( u_i(t) = U_i e^{i\omega t} \) into the differential equations gives the following matrix equation, where \(\alpha = 2k - m\omega^2\):

$$ \begin{vmatrix} \alpha - k & -k & 0 & \dots & 0 & 0 \\ -k & \alpha & -k & \dots & 0 & 0 \\ 0 & -k & \alpha & \dots & 0 & 0 \\ \vdots & \vdots & \vdots & \star & \vdots & \vdots \\ 0 & 0 & 0 & \dots & \alpha & -k \\ 0 & 0 & 0 & \dots & -k & \alpha - k \end{vmatrix} \begin{vmatrix} U_1 \\ U_2 \\ U_3 \\ \vdots \\ U_{n-1} \\ U_n \end{vmatrix} = \begin{vmatrix} C \\ 0 \\ 0 \\ \vdots \\ 0 \\ 0 \end{vmatrix} $$

2. Live System Simulation

Driving force applied to Mass 1. The animation below accurately scales with the calculated displacement amplitudes (\(U_i\)).

3. Generated System Matrix (A)

The dynamically calculated \( n \times n \) tridiagonal matrix based on your inputs. Values are formatted to 2 decimal places.