diff eq of mechanical system w/ friction with zero input.... I was wondering how we get around modeling mechanical systems with frictional forces (forces proportional to normal force and in the direction opposite of the motion) and the external force is zero. So, take for example a second order differential equation modeling spring/mass/damper/friction system with a forced response of zero. the spring force is proportional to displacement, the dampening force is proportional to velocity, and the friction is proportional to weight. The differential equation just doesn't work for situations where the forced response is zero. The input has to equal the frictional force, or else the solution of the equation implies that the system will move out of equilibrium due to the frictional force and oscillate, which is ludicrous. Any explanations? What am I overlooking? The inverse laplace transform of the s-domain solution is the equation of the motion for this system with a frictional force present. the s-domain solution for the displacement I found is -K1/[ms^2 + K2s + K1] where K1 is hooke's constant, K2 is the damping ratio, and K3 is the frictional force, which is a constant. Can someone find the inverse of this? I looked through some tables and couldn't find the solution. So, I guess my question is how to we model this system so it takes friction into account, but will allow for a forced response of zero? I want x(t) to be zero for when f(t) is zero, even with the friction force in the DE. Also, I want the frictional force to switch directions as the oscillating spring switches directions. Say, my input is a constant for some time and then when the spring force equals the input force the input force drops to zero (I let go of the mass). I want to frictional force to maintain in the direction opposite of the motion.... :yuck: Also, the damping force works fine because it is proportional to the velocity. so if the block direction switches direction, the damping force also switches direction. yay. Also, if the initial position is zero, then even if the force is zero the x(t) = 0, since again, damping force is proportional to velocity. Frictional force, however, isn't, and I do not understand how to model it correctly. MAybe this is why my control systems book didn't mention frictional force and skipped ahead to dampening force.