# Equation of Motion for 2 DOF spring damper system

Please guide me towards the "differential equation of motion" for the following 2 DOF Spring-damper system. And furthermore, if above system is in a uniform speed rotating frame, then what can be the effect on this system?

Thank you very much.

## Answers and Replies

Astronuc
Staff Emeritus
Science Advisor
Please write the equation for one dof.

Does gravity g apply to y-direction?

What force must be considered in a rotating body?

Thank you for your reply.

m*(d2x/dt2)+c*(dx/dt)+k*x = 0 is the equation for one axis.

No effect of gravity is considered here.

The whole frame is rotating at an uniform angular velocity.

If you can provide equation for static (i-e. not rotating) case, that's also fine.

Correct me if I'm wrong.

Here are the motion equations;

mx'' + c1x' + k1x' = 0 (Not forced)
my'' + c2y' + k2y' = 0 (Not forced)

Suppose the frame is rotated thru angle q,

In this case, the mass center position with respect to the frame F is

Xnew = x*cos(q) - y*sin(q)
Ynew = y*cos(q) + x*sin(q)

You can simply differentiate Xnew and Ynew. Once and twice, then replace in motion equations

Xnew' = cos(q)*(x'-yq*') - sin(q)*(y'+xq*')
Xnew'' = x''*cos(q) - 2*x'*q'*sin(q) - x*q''*sin(q) - x*q'*q'*cos(q)
-y''*sin(q) - 2*y'*q'*cos(q) - y*q''*cos(q) + y*q'*q'*sin(q)

Note that q''=0. Eliminate some of the terms above and do the same thing for y-axis