Question 5.1.2: Vector-Matrix Form of a Single-Mass Model Express the mass-s...

System Dynamics

Vector-Matrix Form of a Single-Mass Model

Express the mass-spring-damper model (5.1.2) and (5.1.3) as a single vector-matrix equation. These equations are

$m\ddot{x} + c\dot{x} + kx = f$ (5.1.1)
$\dot{x}_{2} = \frac{1}{m} ( f − kx_{1} − cx_{2})$ (5.1.3)

\dot{x}_{1} = x_{2}

x_{2} = \frac{1}{m} f(t)  −  \frac{k}{m} x_{1}  −  \frac{c}{m} x_{2}

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The equations can be written as one equation as follows:

\begin{bmatrix} \dot{x}_{1} \\\dot{x}_{2} \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ – \frac{k}{m} & – \frac{c}{m} \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} + \begin{bmatrix} 0 \\ \frac{1}{m} \end{bmatrix} f(t)

In compact form this is
$\dot{x} = Ax + B f (t)$

A = \begin{bmatrix} 0 & 1 \\ – \frac{k}{m} & – \frac{c}{m} \end{bmatrix}         B = \begin{bmatrix} 0 \\ \frac{1}{m} \end{bmatrix}      x = \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}

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