4.2.2 Vector Arithmetic in Mass Spring Chain

Types: N/A
Examples: N/A
Constructions: 4.2.3 Hooke's Law Data Analysis
Generalizations: N/A

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

Vector Arithmetic in Mass-Spring Chain

Recall our 3.1.2 Vector Model of Mass-Spring Chain. How can we write our displacement vector-valued function $\vec{u} (t)$ using 4.2 Column Vector Addition?

Solution. Let ${\vec{x}}_{0}$ be our mass positions at equilibrium.

{\vec{x}}_{0} = [\begin{matrix} x_{1} \\ x_{2} \end{matrix}]

Let $\vec{x} (t)$ be our mass positions at time $t$ .

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

To find displacement, we subtract the position at time $t$ from our initial position. Thus,

\begin{aligned} \vec{u} (t) = [\begin{array}{c} u_{1} (t) \\ u_{2} (t) \end{array}] & = [\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}

Remark. The operation we used is column vector subtraction which stems from column vector addition.
$⧫$