I am trying to figure out how to come up with a transfer function for a spring mass damper with variable damping. The problem is when I try to come up with a transfer function for my equation I get a convolution but not sure how to handle this with a variable input. Is there any way for me to linearize the system so that I can get a transfer function for it?
$$ m\ddot{x}=k(w-x)+u(\dot{w}-\dot{x}) $$
Where w is the displacement of the road surface and u is the variable damping rate. I appreciate the help!