# Mechanical system modeling

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

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

• In the diagram the damper force is $D_1 \dot{x}_1$ but in the equations you use $Ba_2 \dot{x}$. And you need to decide and show which way is positive displacements & forces and which way negative. Feb 10, 2014 at 1:31
• If spring $K_2$ is attached to mass $m_2$ on one end why isn't appear in the equations of motion for mass 2? Think about what elements apply forces to each isolated mass using two Free Body Diagrams Feb 10, 2014 at 1:33

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.