Mechanical system modeling

Question

I have a mechanical system and I need to model in differential equations

I tried to model the mass $M_1$ and got this

$m_1\displaystyle\frac{d^2\,x}{dt^2}=F-K_1(x_2 - x_1)-Ba_1\displaystyle\frac{dx}{dt}.$

$F$ is the force.

$K_1(x_2 - x_1)$ spring is multiplied by subtracting the distance of the spring and damper.

$Ba_1\displaystyle\frac{dx}{dt}$ is the force of the damper.

I don't understand how to model the mass on the right (D2). I tried, but I need help

$m_2\displaystyle\frac{d^2\,x}{dt^2}=-K_1(x_2 - x_1)-Ba_2\displaystyle\frac{dx}{dt} $.

Answer 1

According to the diagram supplied, and assuming that positive motion and forces point to the right then for a positive displacement of the mass on the left $x_1$ the forces acting upon it are

$$ -F - k_1 x_1 - d_1 \dot{x}_1 - k_2 (x_1 - x_2) = m_1 \ddot{x}_1 $$

similarly for mass on the right

$$ - k_2 (x_2-x_1) - d_2 \dot{x}_2 = m_2 \ddot{x}_2 $$

Your job is to fill in the coefficient matrices such that

$$ \begin{bmatrix} m_1 & 0 \\ 0 & m_2 \end{bmatrix} \begin{pmatrix} \ddot{x}_1 \\ \ddot{x}_2 \end{pmatrix} =\begin{bmatrix} \cdots &\cdots \\ \cdots & \cdots \end{bmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} +\begin{bmatrix} \cdots &\cdots \\ \cdots & \cdots \end{bmatrix} \begin{pmatrix} \dot{x}_1 \\ \dot{x}_2 \end{pmatrix} + \begin{pmatrix} -F \\ 0 \end{pmatrix} $$

allowing you to form a well defined system of differential equations.