# Damping and friction in syringe equation of motion

mamech
Hello Everyone
I want to model forces affecting on syringe plunger , but I do not know how to calculate terms like friction and damping coefficient.
What I imagine is that : F_driving = ma + cv + f ----------------(1)
where:
f: friction
c: coefficient of viscous damping
m: mass of plunger (is n't it?)
v and a : velocity and acceleration respectively

In a real setup, I wanted to determine c and f, I used equation (1) to have 2 equations at different F_driving values (by hanging different masses at pluger tip) , and I waited until I reach steady state of speed (i.e. ma=0), so by solving the 2 equations, I got both c and f.

I have several questions in this regard:
1- Is equation 1 is correct? or did I miss any term?
2- Is the approach used to get f and c is correct?
3- I know that friction is not affected by surface area, but how to integrate this rule with the common sense, where you find more resistance in syringe if the pluger is longer or bigger in diameter (i.e surface area is bigger)?

Thanks

Gold Member
Is this syringe ejecting water or some other incompressible liquid?

Gold Member
I would try ignoring the initial acceleration of the plunger for the time being, because that requires analysis of unsteady flow, I think this is going to be complicated enough for the time being.

If (1) is the FBD on the plunger, the ##cv## term is really a pressure ##PA## term (acting in the opposite the direction of motion) term that will ultimately be a nonlinear function of the fluid jet velocity and a host of other viscous friction parameters, like the viscosity of the liquid, tube diameter, and tube length, and Reynolds number.

It would also be a good idea to (at least initially - depending on how deep you wish dive) operate under the assumption that viscous friction in the body of the syringe is negligible under the assumption ##v_b \ll v_{nozz}##. I say that because the viscous friction in that section would have dependence on the position of the plunger (through the volume of fluid remaining in the syringe that it is pushing on), under steady flow, it's probably manageable, but I'd hold off a moment until a base model is worked out.

I believe (3) is related to the normal force, and the stress induced in compressing a certain volume of plunger material to make the seal. If there is more material that needs compressed the stress increases, and the normal force increases, thus increasing the frictional forces.

As far as (2) goes, I'm not sure yet. In theory if your final model has two unknows, then sure. But with the variations you are talking about different syringe characteristics, I don't think you are going to get there in two equations, two unknowns.

Last edited:
mamech and Lnewqban
mamech
Is this syringe ejecting water or some other incompressible liquid?
For simplification you can consider it water for the time being.

For comment (1), I agree, and I think all are constants, while only flow rate increase affects reynolds number, and hence affects the viscous damping coefficient.
This is also an interesting point that I missed. so when weh talk about damping here, we have 2 velocities to talk about, the one at nozzel and the other at syringe body, and it is logical that damping at nozzel is the dominant part, so all calculations should be referred to it.
but does this means, that I should in equation one substitute the velocity with its value on nozzel, to be representative for the major damping source?

For comment (3), I understand what you want to say, but for example, we know that wider car tyres gives more stability, and snow shoes have bigger contact area with ground than running shoes. If area does not change anything in regard to friction force, then why such applications exist, and work efficiently?

Gold Member
For simplification you can consider it water for the time being.

For comment (1), I agree, and I think all are constants, while only flow rate increase affects reynolds number, and hence affects the viscous damping coefficient.
This is also an interesting point that I missed. so when weh talk about damping here, we have 2 velocities to talk about, the one at nozzel and the other at syringe body, and it is logical that damping at nozzel is the dominant part, so all calculations should be referred to it.
but does this means, that I should in equation one substitute the velocity with its value on nozzel, to be representative for the major damping source?

Yeah, I believe the main viscous force will be in the nozzle, and velocities relating such forces should refer to nozzle flow velocity. The plunger velocity (which is what I assume you intend to measure) is related to nozzle velocity through continuity ## A_1v_1 = A_2 v_2##.

For comment (3), I understand what you want to say, but for example, we know that wider car tyres gives more stability, and snow shoes have bigger contact area with ground than running shoes. If area does not change anything in regard to friction force, then why such applications exist, and work efficiently?

The snowshoes aren't wide for traction so much, they are wide to increase the volume of snow under foot, such as to minimize the stress compressing the snow, and stop you from sinking.

I believe the standard physics model for static friction "##f_r = \mu N ##" fails to accurately capture the effect of surface area on traction - it is apparently a "freshman physics" simplification that is very close for some materials and in the ballpark for others. Friction is actually microscopic, Coulombic Forces of molecular scale interactions between the materials...It is in that regard that surface area comes into play, and makes sense that more surface area, means more bonds, higher friction force, which is why rear tires of top fuel drag cars are wide, flat, and relatively soft so the powertrain can deliver massive amounts of torque without the tires slipping while simultaneously minimizing the weight ( the normal force ) of the vehicle so that acceleration can be maximized.

As far as a model for that goes, I'm ignorant of one.

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mamech