newtonian mechanics - Friction forces and sliding slabs

I have 2 questions, one generalizing the other.

Question 1: Suppose we have 2 slabs resting horizontally on a table. Assume there is friction between the 2 slabs as well as between the bottom slab and the table and that all friction coefficients are different. Now we apply a horizontal force to the top slab. How do we figure out the direction of movement of the bottom slab?

Question 2: We now have a stack of n slabs resting horizontally on the table. All surfaces in contact have friction, and all friction coefficients are different. If we apply a horizontal force to the top slab, how can we predict the direction of movement of each slab in the stack?

For the first question, I am guessing that depending on the amount of force and the values of the friction coefficients, there can be multiple scenarios: the slabs won't move till the force overcomes the first static friction coefficient, then they might move together, then one might move in one direction, and the other... well, how does the bottom slab move exactly? That's where I am confused. It seems I have to know the direction of movement to set the sign of the friction force between the bottom slab and the table correctly, but I am not sure how to establish the force transmitted by the top slab to the bottom one. Is it just the friction force between top and bottom slab?

For the second question, I'd gladly apply the same method as for the first question repetitively, but lo and behold, I haven't solved the first question yet...

This is not for a "homework" and I am not a student trying to get his/her homework answered :-) Thanks!

Answer

So you want the formal answer to question 2? Read on:

Lets say we have $k$ blocks, numbered $i=1 \ldots k$ with 1 on the bottom and $k$ on the top. The top block has an applied force $\mathcal{P}$ and each block has mass $m_i$ and friction coefficient with the previous block (or the ground) $\mu_i$. Also the movement of each block is characterized by the acceleration $\ddot{x}_i$. In matrix form the above define

$$ P=\begin{pmatrix}0\\ 0\\ \vdots\\ 0\\ \mathcal{P} \end{pmatrix} $$ $$ m=\begin{bmatrix}m_{1}\\ & m_{2}\\ & & \ddots\\ & & & m_{k-1}\\ & & & & m_{k} \end{bmatrix} $$ $$ \mu=\begin{bmatrix}\mu_{1}\\ & \mu_{2}\\ & & \ddots\\ & & & \mu_{k-1}\\ & & & & \mu_{k} \end{bmatrix} $$ $$ \ddot{x}=\begin{pmatrix}\ddot{x}_{1}\\ \ddot{x}_{2}\\ \vdots\\ \ddot{x}_{k-1}\\ \ddot{x}_{k} \end{pmatrix} $$

The weight on each block is $m_i g$ and the contact force with the previous block (or the ground) is $N_i$. Also the friction limit is $F_i \leq \mu_i N_i$. In matrix form the above is

$$ N=\begin{pmatrix}N_{1}\\ N_{2}\\ \vdots\\ N_{k-1}\\ N_{k} \end{pmatrix} $$

$$ F \leq \begin{bmatrix}\mu_{1}\\ & \mu_{2}\\ & & \ddots\\ & & & \mu_{k-1}\\ & & & & \mu_{k} \end{bmatrix}\begin{pmatrix}N_{1}\\ N_{2}\\ \vdots\\ N_{k-1}\\ N_{k} \end{pmatrix}=\begin{pmatrix}\mu_{1}N_{1}\\ \mu_{2}N_{2}\\ \vdots\\ \mu_{k-1}N_{k-1}\\ \mu_{k}N_{k} \end{pmatrix} $$

Why do we need all this? To to make the equation of motion for the $i$-th block, which is $P_i - F_i + F_{i+1} = m_i \ddot{x}_i $

Look at the free body diagram above. By convention the i-th friction opposes the motion which is to the right. The friction from the above block is reacted upon this block and applied to the left. That is why the sum of the fores is $P_i + F_{i+1} - F_i$.

The balance in matrix form, using an adjacency matrix is

$$ A=\begin{bmatrix}1 & -1\\ & 1 & -1\\ & & \ddots & \ddots\\ & & & 1 & -1\\ & & & & 1 \end{bmatrix} $$ $$ P-A\, F=m\ddot{x} $$

which expands out to

$$\begin{pmatrix}0\\ 0\\ \vdots\\ 0\\ \mathcal{P} \end{pmatrix}+\begin{pmatrix}F_{2}-F_{1}\\ F_{3}-F_{2}\\ \vdots\\ F_{k}-F_{k-1}\\ -F_{k} \end{pmatrix}=\begin{pmatrix}m_{1}\ddot{x}_{1}\\ m_{2}\ddot{x}_{2}\\ \vdots\\ m_{k-1}\ddot{x}_{k-1}\\ m_{k}\ddot{x}_{k} \end{pmatrix}$$

Now the contact normal force is derived from the blocks above it with

$$ A\,N = m\,g $$ $$ N = A^{-1} m\,g $$ $$ \begin{pmatrix}N_{1}\\ N_{2}\\ \vdots\\ N_{k-1}\\ N_{k} \end{pmatrix}=\begin{bmatrix}1 & 1 & 1 & 1 & 1\\ & 1 & 1 & 1 & 1\\ & & \ddots & \vdots & \vdots\\ & & & 1 & 1\\ & & & & 1 \end{bmatrix}\begin{pmatrix}m_{1}g\\ m_{2}g\\ \vdots\\ m_{k-1}g\\ m_{k}g \end{pmatrix} $$

So all together $$ P - \left( A\,\mu A^{-1}\right) m\, g=m\ddot{x} $$

or with $ \mu_{SYS}=A\,\mu A^{-1} $

$$ \mu_{SYS}=\begin{bmatrix}1 & -1\\ & 1 & -1\\ & & \ddots & \ddots\\ & & & 1 & -1\\ & & & & 1 \end{bmatrix}\begin{bmatrix}\mu_{1}\\ & \mu_{2}\\ & & \ddots\\ & & & \mu_{k-1}\\ & & & & \mu_{k} \end{bmatrix}\begin{bmatrix}1 & 1 & 1 & 1 & 1\\ & 1 & 1 & 1 & 1\\ & & \ddots & \vdots & \vdots\\ & & & 1 & 1\\ & & & & 1 \end{bmatrix} \\ \mu_{SYS}=\begin{bmatrix}\mu_{1} & \mu_{1}-\mu_{2} & \cdots & \mu_{1}-\mu_{2} & \mu_{1}-\mu_{2}\\ & \mu_{2} & \cdots & \mu_{2}-\mu_{3} & \mu_{2}-\mu_{3}\\ & & \ddots & \vdots & \vdots\\ & & & \mu_{k-1} & \mu_{k-1}-\mu_{k}\\ & & & & \mu_{k} \end{bmatrix} $$

$$ P -\mu_{SYS} m\, g=m\ddot{x} $$ $$ \ddot{x} = m^{-1} \left(P-\mu_{SYS} m\, g \right) $$

So this is the motion once with have slipping. We need to reverse the equations and find the traction required when $\ddot{x}=0$ which ends up being

$$ \mu_i \geq \frac{\mathcal{P}}{g (\sum_{j=i}^k m_j)} $$

When the above is not satisfied the contact is slipping. Otherwise system will have $\ddot{x}_i=0$ for when the contact sticks.

Block Matrix Solution

Here are the steps needed to solve the above system

1. Stick all contacts with $\ddot{x}=0$ and find the friction needed $F^{\star}=A^{-1}P$. For example $$F^{\star}=\begin{bmatrix}1 & 1 & \cdots & 1 & 1\\ & 1 & \cdots & 1 & 1\\ & & \ddots & \vdots & \vdots\\ & & & 1 & 1\\ & & & & 1 \end{bmatrix}\begin{pmatrix}0\\ 0\\ \vdots\\ 0\\ \mathcal{P} \end{pmatrix}=\begin{pmatrix}\mathcal{P}\\ \mathcal{P}\\ \vdots\\ \mathcal{P}\\ \mathcal{P} \end{pmatrix}$$


2. Compose the system mass matrix $M=A^{-1}m$ such that the horizontal equations of motion are $\boxed{F^{\star}=M\ddot{x}+F}$


3. Compare friction needed to available traction with $F^{\star}<\mu N$. Construct two projection matrices $T$ and $U$ with $k$ rows and values as follows: For each block $i$ that is sliding add a column to $U$ with the i-th row element equal to 1 and all others 0. For each block $i$ that is sticking add a column to $T$ with the i-th row element equal to 1 and all others 0. For example if only the last element (top) slides then $$ \begin{aligned} T&=\begin{bmatrix}1\\ & 1\\ & & \ddots\\ & & & 1\\ & & & 0 \end{bmatrix}&U&=\begin{bmatrix}0\\ 0\\ \vdots\\ 0\\ 1 \end{bmatrix} \end{aligned}$$


4. Define the known motions (sticking blocks) with $T^{\top}\ddot{x}=0$ and the known friction (sliding blocks) with $f=U^{\top}F=U^{\top}\mu N$. With the example above then $$\begin{aligned} \begin{pmatrix}0\\ 0\\ \vdots\\ 0 \end{pmatrix}&=\begin{bmatrix}1\\ & 1\\ & & \ddots\\ & & & 1\\ & & & 0 \end{bmatrix}^{\top}\begin{pmatrix}\ddot{x}_{1}\\ \ddot{x}_{2}\\ \vdots\\ \ddot{x}_{k-1}\\ \ddot{x}_{k} \end{pmatrix}=\begin{pmatrix}\ddot{x}_{1}\\ \ddot{x}_{2}\\ \vdots\\ \ddot{x}_{k-1} \end{pmatrix}\\f&=\begin{bmatrix}0\\ 0\\ \vdots\\ 0\\ 1 \end{bmatrix}^{\top}\begin{pmatrix}\mu_{1}N_{1}\\ \mu_{2}N_{2}\\ \vdots\\ \mu_{k-1}N_{k-1}\\ \mu_{k}N_{k} \end{pmatrix}=\begin{pmatrix}\mu_{k}N_{k}\end{pmatrix} \end{aligned}$$


5. Define the unknown motions vector $a$ and unknown forces vector $R$ such that the block motion is $\ddot{x}=U\, a$ and the block friction $F=T\, R+M\, U\left(U^{\top}M\, U\right)^{-1}f$. Note that $U^{\top}F=f$ and $T^{\top}M^{-1}F=\left(T^{\top}M^{-1}T\right)\, R$.



6. The horizontal equations of motion are $\boxed{ F^{\star}=T\, R+M\, U\left(a+\left(U^{\top}M\, U\right)^{-1}f\right)}$ with $R$ and $a$ as unknowns.


7. Project to the sliding blocks with $U^{\top}F^{\star}=U^{\top}M\, U\left(a+\left(U^{\top}M\, U\right)^{-1}f\right)$ } $\boxed{a=\left(U^{\top}M\, U\right)^{-1}\left(U^{\top}F^{\star}-f\right)}$


8. Project to the sticking blocks with $T^{\top}M^{-1}F^{\star}=\left(T^{\top}M^{-1}T\right)\, R$ } $\boxed{R=\left(T^{\top}M^{-1}T\right)^{-1}T^{\top}M^{-1}F^{\star}}$


9. Back substitute the projections to get $\ddot{x}=U\, a$ and $F=F^\star - M \ddot{x}$.