How to Derive Differential Equations for a Second Order System?

[MarkD88]

I have attached an image of a question I am trying to do, I want to find the differential equations that describe the second order system in the image.

I know for a spring, potential energy = 1/2.K.x (where k is the spring constant, and x is the distance the spring is stretched).

I know that damper dissipation is = 1/2.B.(dx/dt) (where B is the damping constant and dx/dt is the rate of change of position with respect to time).

My solution:

I try to build a free body diagram for each mass on its own:

I have showed a snapshot of my attempt for the D.E. describing the motion of m1.

If any of you guys could tell me if I am correct or you have a handy way of doing these types of questions please feel free to respond.

Thank you

 

[BvU]

Hello Mark, welcome to PF !

For some reason the template has disappeared. Pity (*). The relevant equation is $\Sigma F = ma$. So you want to sort out the signs a little more carefully (of the forces; $-k$ is not a force, nor is $-B$ ).(*) Its use is mandatory in PF, for very good reasons; see the guidelines.

 

[MarkD88]

Hi BvU,

Apologies about the template, I will not post without a template again.

My notation there was a little careless, I did not mean to label K and B as forces, I was trying to imply that there is a force acting on mass m1 due to both the spring and the damper.

Spring: k.x1
Damper: B.(x2dot-x1dot)

And that these two forces oppose the motion of mass m1 from left to right.

 

[BvU]

And that these two forces oppose the motion of mass m1 from left to right

is good. And when you state that, you should also write something like $m_1\ddot x_1 = -kx_1$ and not e.g. $m_1\ddot x_1 -kx_1 = 0$