# Simultaneous: static friction and acceleration

I need help thinking about two events happening simultaneously.

I have a heavy block on a surface with static and kinetic coefficients of friction, though they are unknown. The block is connected to a linear actuator that will pull with a constant velocity (not varying force) when powered. Between the block and the linear actuator is a load cell that will read real-time force, so the load cell is in tension. Let's assume the load cell is in perfect alignment with the actuator and the block so it is not being loaded in a way to give false readings and its sampling frequency is appropriate. The purpose of this experiment is to measure the force to overcome static friction between the block and the surface.

Ok, so I power the linear actuator and it moves from rest to its rated constant velocity. The load cell gives readings: ramps up quickly to a peak then decreases and stabilizes when the block is sliding at a constant velocity. Here's my question: I didn't think that the velocity of the linear actuator had anything to do with the peak load cell reading, but I think it does. I think I'm overcoming static friction at the instant I'm also accelerating the block from rest to the rated velocity of the linear actuator. Would these "combined" forces happening pretty much simultaneously be reflected by the higher than expected force readings on my load cell?

Suppose we ignore the friction and take the surface to be frictionless - we can add back the friction later. If the block velocity as a function of time is $v_b(t)$ then the acceleration of the block is:

$$a_b = \frac{dv_b(t)}{dt}$$

and from Newton's second law the force on the block is:

$$F_b = ma_b = m\frac{dv_b(t)}{dt}$$

So the initial force you measure does indeed depend on how your actuator goes from rest to some velocity $v_a$. In fact it's worse than that because the load cell and the coupling between the actuator and the block will have some compliance so $v_b(t) \ne v_a(t)$. Unless you know the compliance of the coupling you can't measure $v_b(t)$ so you can't predict the inertial force $F_b$. Since you can't predict $F_b$ you can't subtract it from the total force and you can't calculate the static friction.

If you want to measure the static friction you need a different actuator that can apply a steadily increasing force until the block moves.

• I think you are correct and have confirmed what I have also been suspecting. Looks like I need a new test set up... :) – A.P. Oct 17 '15 at 17:15

Keep in mind that the friction models, $F_s\le F_{smax}=\mu_s\ F_{normal}$ and $F_k=\mu_k\ F_{normal}$ are first-order, experimentally-based models that work fairly well in general. They are not perfect, so getting a spread of $F_{smax}$ values is not surprising.

What you should do is obtain a range of $F_{smax}$ values for a variety of actuator speeds and a variety of block masses. This will tell you if there is some first-order clustering around a central value. Then you will also have some speed vs normal force vs $F_{smax}$ data that you can investigate.

Based on what I've done in a class-based lab, pushing a force sensor by hand against a stationary block, the most "reliable" data come from vary slow increases of the pushing force.

• I agree with both yours and John's comments. I think the force reading I am getting is both influenced by accelerating the block and the fact that static friction is a complex phenomenon and not first order, which may be influencing my results. We all agree that a constant linear actuator that does not vary force is likely not the best way to assess what I'm after. – A.P. Oct 17 '15 at 17:17
• Upvotes are appropriate for agreeing with answers. :) – Bill N Oct 17 '15 at 17:45
• Mr. Downvote, please comment so I can address the problem. – Bill N Nov 4 '15 at 21:18