Types: N/A
Examples: N/A
Constructions: 4.2.2 Vector Arithmetic in Mass Spring Chain
Generalizations: 3.1 Column Vector, 00 Major Problems in Applied Linear Algebra

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Vector Model of Mass-Spring Chain

In Lesson 5, we will model coupled mass-spring chains using linear systems of equations. For now, let's construct a vector model for a 2-mass, 3-spring chain to encode the positions of our masses at a given point in time. In other words, let's create a model to store our recorded data.

Equilibrium

Let's create a vector model to capture the equilibrium position of each mass in a two mass, three spring chain when no external forces are applied. Consider

\begin{aligned} {\vec{x}}_{0} = {[\begin{array}{c} x_{1} \\ x_{2} \end{array}]}_{2 \times 1} & {\vec{x}}_{0} \in R^{2 \times 1} \end{aligned}

Each entry of ${\vec{x}}_{0}$ is a constant (real number) where

$x_{1}$ is the position measurement (in meters) of center of mass $m_{1}$ when system is at rest.
$x_{2}$ is the position measurement (in meters) of center of mass $m_{2}$ when system is at rest.

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Spring Mass Chain at equilibrium
with NO external force (t=0 sec)

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Embedded Files

3b1137dc3488130f0426281066cc4cdb86150ca1: $x_{1}$
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b4c673079ee8391431f52a861441e23c282935ea: $x_{2}$
b37230525c3687593a13ef4286596359b835cb6a: $m_{1}$
67adcbf0e450fd865247abdf223b5fdbbfb6b193: $m_{2}$
30ce668f797c9c1a4864fecddd89d649f36e9bbe: $x_{2} (t)$
4220b76d93da23ed409042a358e0ba65cb445c12: $m_{1}$
bd3e400664b2f1c56c5dd7b6a03ec869ffd8a086: $m_{2}$
dbbee70a8808cc8d9dafe49327e52eaacb658d63: $u_{1} (t)$
49e1fa22f8b8a6dd52e6b8fa147915a9b6ce3144: $u_{2} (t)$
eb390ed816872c4ee1e1a3cab7b6270bc7c313f8: [[Pasted Image 20240404224753_081.png]]

External Force

Now, we create a vector-valued position function to capture the system with external force applied.

\vec{x} (t) = {[\begin{matrix} x_{1} (t) \\ x_{2} (t) \end{matrix}]}_{2 \times 1}

We write $\vec{x} (t)$ as a function

\vec{x} : \underset{domain of \vec{x} (t)}{\underset{⏟}{[t_{0}, T)}} \to R^{2 \times 1}

$t_{0}$ is the initial time when we start recording position measurement in our system
$T$ is the final time when we stop recording position data

Each entry $\vec{x} (t)$ encodes the position function describing the location of the center of mass $i$ along metric ruler at time $t$ :

$x_{1} (t)$ is the position of center of mass $i$ (in meters) at time $t$ (in seconds)
$x_{2} (t)$ is the position of center of mass $i$ (in meters) at time $t$ (in seconds)

Displacement

Now lets create a function to measure the displacement (change in position) of each mass:

\begin{aligned} \vec{u} (\overset{scalar input}{t}) & = {[\begin{array}{c} u_{1} (t) \\ u_{2} (t) \end{array}]}_{2 \times 1 \leftarrow vector valued output} \\ = [\begin{array}{c} x_{1} (t) - x_{1} \\ x_{2} (t) - x_{2} \end{array}] \\ = [\begin{array}{c} x_{1} (t) \\ x_{2} (t) \end{array}] - [\begin{array}{c} x_{1} \\ x_{2} \end{array}] \\ = \vec{x} (t) - {\vec{x}}_{0} \end{aligned}

Notice that our displacement function $\vec{u} (t)$ has vector-valued output.

\vec{u} : \underset{Domain of \vec{u} (t)}{\underset{⏟}{[t_{0}, T)}} \to R^{2 \times 1}

Our displacement function's domain is the same domain as $\vec{x} (t)$ . Moreover, we know:

\begin{aligned} u_{1} (t) & = x_{1} (t) - x_{1} \\ = displacement of center of mass 1 from equilibrium at time t \\ u_{2} (t) & = x_{2} (t) - x_{2} \\ = displacement of center of mass 2 from equilibrium at time t \end{aligned}

If we know masses $m_{1}$ and $m_{2}$ , and spring constants $k_{1}$ , $k_{2}$ , and $k_{3}$ , can we generate a model $\vec{u} (t)$ that accurately predicts behavior of measured displacement data for these masses? In other orders, can we mathematize this problem?