# Diff eq of mechanical system w/ friction with zero input

1. Sep 3, 2006

### leright

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.

Last edited: Sep 3, 2006
2. Sep 3, 2006

### leright

maybe this is too complicated of a problem and that is why it seems to be ignored by all textbooks. meh.

3. Sep 3, 2006

### leright

anyone??? This thread was viewed by 15 physics experts and no insight? :p

4. Sep 3, 2006

### Claude Bile

Use imaginary components in the spring constants to represent the friction (or attenuation) of the oscillation.

Claude.

5. Sep 3, 2006

### leright

I'm not sure I follow.

6. Sep 3, 2006

### leright

Could someone elaborate on Claude's comment? It may be beyond my mathematical experience.

7. Sep 3, 2006

### pseudovector

Claude meant you should use phasors to describe the oscillation. The phasor for a frictionless spring would be x=A*exp(i*w*t) (A- amplitude, w - angular frequency, t- time, i- sqrt(-1)). Notice that when you take the real part of a phasor, you get back the more familiar form of the displacement
Re(x)=A*cos(w*t) according to De Moivre's theorem.
When you put a complex frequency w+i*g instead of real one you get the equation of motion for an oscillator under the influence of friction (aka decaying oscillation).
x=A*exp(i*(w+ig)*t)=A*exp(-g*t)*exp(i*w*t)
Re(x)=A*exp(-g*t)*cos(w*t)

8. Sep 3, 2006

### leright

ok, so let's take the DE for a spring/mass/damper system with a constant frictional force, and apply a generic external input force to the system. The DE is,

m*(second derivative of displacement) + (damping ratio)*(first derivative of displacement) + (spring constant)*(displacement) = input force.

Now, I want to modify this DE so that it accounts for a constant frictional force. Can someone show me how this is done?

9. Sep 3, 2006

### leright

edit: sorry for the double post.

10. Sep 3, 2006

### leright

also, pseudovector, that gives me a better understanding of why cosine functions with imaginary arguments are really damped consines.

interesting.

11. Sep 3, 2006

### leright

Also, say for instance we have a constant input force and pull on the block until the spring force balances the input force. We now have equilibrium and the block no longer moves. Will the frictional force modification mentioned above work in this case? The frictional force should be zero in equilibrium, but the spring force is NOT zero. Is this a flaw in our model?

12. Sep 5, 2006

### SGT

You must model your friction force as $$F_f = -K_3\cdot sign(v)$$.
Where the function sign(x) is
• 1 if x > 0
• 0 if x = 0
• -1 if x < 0

13. Sep 5, 2006

### leright

damn, that means my system is non-linear and cannot be laplace transformed, so I cannot find the transfer function.

14. Sep 12, 2006

### SGT

Yes, dry friction is a nonlinear effect.