Modeling of multi-body systems with freeplay and friction (kinetic and static) provide strong non-linear inclusion forms or forms containing variable-structure ordinary differential equations with algebraic constrains.. Such models are difficult for analysis and simulation. A quest of more “friendly” methods of modeling has ever been an attractive challenge.

In cases of steering mechanisms, the freeplay and friction actions can be expressed by piecewise linear dynamical models with piecewise linear characteristics (Fig. 2).

The characteristics presented in Fig. 2 can be described analytically as:

$${F}_{S}=k\bullet luz\left(z,{z}_{0}\right)$$

1

$${F}_{T}=\left\{\begin{array}{ccc}C tar\left(\dot{z},\frac{{F}_{T0}}{C}\right)& if& \dot{z}\ne 0\\ F-luz\left(F,{F}_{T0}\right)& if& \dot{z}=0\end{array}\right.$$

2

where *luz(...)* and *tar(…)* are piecewise linear projections (Fig. 3) defined below (here a ≥ 0):

$$luz\left(x,a\right)=x+\frac{\left|x-\left.a\right|-\left|x+a\right|\right.}{2}$$

3

\(tar\left(x,a\right)=x+a\cdot sgh\left(x\right)\) where \(sgh\left(x\right)=\left\{\begin{array}{ccc}-1& if& x<0\\ {s}^{*}\in \left[-\text{1,1}\right]& if& x=0\\ 1& if& x>0\end{array}\right.\) (4)

The luz(…) and tar(…) are reversible projections. They have a lot of mathematical properties which create their common mathematical apparatus [8]. Examples formulas (here all parameters ≥ 0):

\(luz\left(x,a\right)=ta{r}^{-1}\left(x,a\right)\) , \(luz\left(x,a\right)=ta{r}^{-1}\left(x,a\right)\), (5, 6)

\(luz\left(-x,a\right)=-luz\left(x,a\right)\) \(tar\left(-x,a\right)=-tar\left(x,a\right)\) (7, 8)

\(k luz\left(x,a\right)=luz\left(k x,k a\right)\) , \(k tar\left(x,a\right)=tar\left(k x,k a\right)\), (9, 10)

\(luz\left(luz\left(x,a\right),b\right)=luz\left(x,a+b\right)\) , \(tar\left(tar\left(x,a\right),b\right)=tar\left(x,a+b\right)\), (11, 12)

$${k}_{1}tar\left(x,{a}_{1}\right)+{k}_{2}tar\left(x,{a}_{2}\right)=\left({k}_{1}+{k}_{2}\right)tar\left(x,\frac{{k}_{1}{a}_{1}+{k}_{2}{a}_{2}}{{k}_{1}+{k}_{2}}\right)$$

13

If \(luz\left(y,b\right)=k luz\left(x-y,a\right)\) then \(luz\left(y,b\right)=\frac{k}{k+1}luz\left(x,a+b\right)\) (14)

If \(0\in y-k tar\left(x,a)\right)\) then \(x=luz\left(\frac{1}{k}y,a\right)\) (15)

The *luz(…)* and *tar(…)* mathematical apparatus is very useful for the synthesis of the piecewise linear mathematical models and their formal parametrically-made reductions. An explanation of this important property is provided below with a simple demonstrative example.

**Example**

Notation:

M1, M2 – masses,

K – stiffness coefficient,

z0 – freeplay parameter,

C – viscous friction coefficient,

FT0 – dry friction parameter,

z – relative displacement,

F1, F2 – acting forces.

For the double-mass system with freeplay and friction (Fig. 4) the starting dynamical model has a compact but inclusion-type form (with unknown formulation for zero speeds).

According to (1) and (2), a starting inclusion-type model is:

$${M}_{1}\cdot {\ddot{z}}_{1}\left(t\right)=\left\{\begin{array}{ccc}-C\cdot tar\left(\left({\dot{z}}_{1}\left(t\right)-{\dot{z}}_{2}\left(t\right)\right),\frac{{F}_{T0}}{C}\right)-K\cdot luz\left(\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right)\right),\left({z}_{1}-{z}_{2}\right)\right)+{F}_{1}\left(t\right)& if& {\dot{z}}_{1}\left(t\right)\ne {\dot{z}}_{2}\left(t\right)\\ -{F}_{T0}\cdot {s}^{*}\left(t\right)-K\cdot luz\left(\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right)\right),\left({z}_{1}-{z}_{2}\right)\right)+{F}_{1}\left(t\right),\text{ }{s}^{*}\left(t\right)\in \left[-\text{1,1}\right]& if& {\dot{z}}_{1}\left(t\right)={\dot{z}}_{2}\left(t\right)\end{array}\right.$$

16

$${M}_{2}\cdot {\ddot{z}}_{2}\left(t\right)=\left\{\begin{array}{ccc}C\cdot tar\left(\left({\dot{z}}_{1}\left(t\right)-{\dot{z}}_{2}\left(t\right)\right),\frac{{F}_{T0}}{C}\right)+K\cdot luz\left(\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right)\right),\left({z}_{1}-{z}_{2}\right)\right)+{F}_{2}\left(t\right)& if& {\dot{z}}_{1}\left(t\right)\ne {\dot{z}}_{2}\left(t\right)\\ {F}_{T0}\cdot {s}^{*}\left(t\right)+K\cdot luz\left(\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right)\right),\left({z}_{1}-{z}_{2}\right)\right)+{F}_{2}\left(t\right),\text{ }{s}^{*}\left(t\right)\in \left[-\text{1,1}\right]& if& {\dot{z}}_{1}\left(t\right)={\dot{z}}_{2}\left(t\right)\end{array}\right.$$

17

Formal calculation of \({s}^{*}\left(t\right)\) (or static friction force \({F}_{T0}\cdot {s}^{*}\left(t\right)\)) is based on the Gauss rule, through the minimization of acceleration energy Q. Here:

$$Q\left(t\right)=\frac{{M}_{1}\cdot {\left({\ddot{z}}_{1}\left(t\right)\right)}^{2}}{2}+\frac{{M}_{1}\cdot {\left({\ddot{z}}_{1}\left(t\right)\right)}^{2}}{2}=\frac{{\left({F}_{11}\left(t\right)-{F}_{T0}\cdot {s}^{*}\left(t\right)\right)}^{2}}{2{M}_{1}}+\frac{{\left({F}_{22}\left(t\right)+{F}_{T0}\cdot {s}^{*}\left(t\right)\right)}^{2}}{2{M}_{2}}=$$

$$=\frac{{M}_{2}{\left({F}_{11}\left(t\right)-{F}_{T0}\cdot {s}^{*}\left(t\right)\right)}^{2}+{M}_{1}{\left({F}_{22}\left(t\right)+{F}_{T0}\cdot {s}^{*}\left(t\right)\right)}^{2}}{2{M}_{1}{M}_{2}}=$$

$$=\frac{\left({M}_{1}+{M}_{2}\right)\cdot {F}_{T0}}{2{M}_{1}{M}_{2}}{\left({s}^{*}\left(t\right)-\frac{{M}_{2}{F}_{11}\left(t\right)-{M}_{1}{F}_{22}\left(t\right)}{\left({M}_{1}+{M}_{2}\right){F}_{T0}}\right)}^{2}+\frac{{\left({F}_{11}\left(t\right)+{F}_{22}\left(t\right)\right)}^{2}}{2{M}_{1}{M}_{2}}$$

18

where

$${F}_{11}\left(t\right)=-K\cdot luz\left(\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right)\right),{\left({z}_{1}-{z}_{2}\right)}_{0}\right)+{F}_{1}\left(t\right)$$

19

$${F}_{22}\left(t\right)=K\cdot luz\left(\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right)\right),{\left({z}_{1}-{z}_{2}\right)}_{0}\right)+{F}_{2}\left(t\right)$$

20

The minimization task:

$${s}^{*}:\text{ }\underset{{s}^{*}}{min}\left(\frac{\left({M}_{1}+{M}_{2}\right){F}_{T0}}{2{M}_{1}{M}_{2}}{\left({s}^{*}-\frac{{M}_{2}{F}_{11}-{M}_{1}{F}_{22}}{\left({M}_{1}+{M}_{2}\right){F}_{T0}}\right)}^{2}+\frac{{\left({F}_{11}+{F}_{22}\right)}^{2}}{2{M}_{1}{M}_{2}}\right)\text{ }\wedge \text{ }{\text{s}}^{*}\in [-\text{1,1}]$$

21

For \({s}^{*}\in [-\text{1,1}]\)the optimal solution is \({s}^{*}=\frac{{M}_{2}{F}_{11}-{M}_{1}{F}_{22}}{\left({M}_{1}+{M}_{2}\right){F}_{T0}}\) (22)

For arbitrary \({F}_{11}\) and \({F}_{22}\), the solution \({s}^{*}\left({F}_{11},{F}_{22}\right)\) must be saturated. Therefore finally:

$${s}^{*}\left(t\right)=\frac{{M}_{2}\cdot {F}_{11}\left(t\right)-{M}_{1}\cdot {F}_{22}\left(t\right)}{\left({M}_{1}+{M}_{2}\right){F}_{T0}}-luz\left(\frac{{M}_{2}\cdot {F}_{11}\left(t\right)-{M}_{1}\cdot {F}_{22}\left(t\right)}{\left({M}_{1}+{M}_{2}\right){F}_{T012}},1\right)$$

23

$${F}_{T0}\cdot {s}^{*}\left(t\right)=\frac{{M}_{2}\cdot {F}_{11}\left(t\right)-{M}_{1}\cdot {F}_{22}\left(t\right)}{{M}_{1}+{M}_{2}}-luz\left(\frac{{M}_{2}\cdot {F}_{11}\left(t\right)-{M}_{1}\cdot {F}_{22}\left(t\right)}{{M}_{1}+{M}_{2}},{F}_{T0}\right)$$

24

So, using the Gauss rule, the model can be presented by variable structure differential equations:

$${M}_{1}\cdot {\ddot{z}}_{1}\left(t\right)=\left\{\begin{array}{ccc}-C\cdot tar\left(\left({\dot{z}}_{1}\left(t\right)-{\dot{z}}_{2}\left(t\right)\right),\frac{{F}_{T0}}{C}\right)-K\cdot luz\left(\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right)\right),{\left({z}_{1}-{z}_{2}\right)}_{0}\right)+{F}_{1}\left(t\right)& if& {\dot{z}}_{1}\left(t\right)\ne {\dot{z}}_{2}\left(t\right)\\ \frac{{M}_{1}}{{M}_{1}+{M}_{2}}\cdot \left({F}_{1}\left(t\right)+{F}_{2}\left(t\right)\right)+luz\left(-K\cdot luz\left({\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right),\left({z}_{1}-{z}_{2}\right)\right)}_{0}\right)+\frac{{M}_{2}\cdot {F}_{1}\left(t\right)-{M}_{1}\cdot {F}_{2}\left(t\right)}{{M}_{1}+{M}_{2}},{F}_{T0}\right)& if& {\dot{z}}_{1}\left(t\right)={\dot{z}}_{2}\left(t\right)\end{array}\right.$$

25

$${M}_{2}\cdot {\ddot{z}}_{2}\left(t\right)=\left\{\begin{array}{ccc}C\cdot tar\left(\left({\dot{z}}_{1}\left(t\right)-{\dot{z}}_{2}\left(t\right)\right),\frac{{F}_{T0}}{C}\right)+K\cdot luz\left(\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right)\right),{\left({z}_{1}-{z}_{2}\right)}_{0}\right)+{F}_{2}\left(t\right)& if& {\dot{z}}_{1}\left(t\right)\ne {\dot{z}}_{2}\left(t\right)\\ \frac{{M}_{2}}{{M}_{1}+{M}_{2}}\cdot \left({F}_{1}\left(t\right)+{F}_{2}\left(t\right)\right)-luz\left(-K\cdot luz\left({\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right),\left({z}_{1}-{z}_{2}\right)\right)}_{0}\right)+\frac{{M}_{2}\cdot {F}_{1}\left(t\right)-{M}_{1}\cdot {F}_{2}\left(t\right)}{{M}_{1}+{M}_{2}},{F}_{T0}\right)& if& {\dot{z}}_{1}\left(t\right)={\dot{z}}_{2}\left(t\right)\end{array}\right.$$

26

The presented model has a clear interpretation:

**Note 1**

When \({\dot{z}}_{1}\left(t\right)={\dot{z}}_{2}\left(t\right)\) and \(-{F}_{T0}\le -K\cdot luz\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right),{\left({z}_{1}-{z}_{2}\right)}_{0}\right)+\frac{{M}_{2}\cdot {F}_{1}\left(t\right)-{M}_{1}\cdot {F}_{2}\left(t\right)}{{M}_{1}+{M}_{2}}\le {F}_{T0}\) ,

then

\(\left({M}_{1}+{M}_{2}\right)\cdot {\ddot{z}}_{1}\left(t\right)={F}_{1}\left(t\right)+{F}_{2}\left(t\right)\) and \(\left({M}_{1}+{M}_{2}\right)\cdot {\ddot{z}}_{2}\left(t\right)={F}_{1}\left(t\right)+{F}_{2}\left(t\right)\) (27, 28)

These equations have identical forms. It means that \({\ddot{z}}_{1}\left(t\right)={\ddot{z}}_{2}\left(t\right)\) (stick state).

**Note 2**

When \({\dot{z}}_{1}\left(t\right)={\dot{z}}_{2}\left(t\right)\) and \(\frac{{M}_{2}\cdot {F}_{1}\left(t\right)-{M}_{1}\cdot {F}_{2}\left(t\right)}{{M}_{1}+{M}_{2}}=0\),

$${M}_{1}\cdot {\ddot{z}}_{1}\left(t\right)=\frac{{M}_{1}}{{M}_{1}+{M}_{2}}\cdot \left({F}_{1}\left(t\right)+{F}_{2}\left(t\right)\right)+luz\left(-K\cdot luz\left({\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right),\left({z}_{1}-{z}_{2}\right)\right)}_{0}\right),{F}_{T0}\right)$$

29

$${M}_{2}\cdot {\ddot{z}}_{2}\left(t\right)=\frac{{M}_{2}}{{M}_{1}+{M}_{2}}\cdot \left({F}_{1}\left(t\right)+{F}_{2}\left(t\right)\right)-luz\left(-K\cdot luz\left({\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right),\left({z}_{1}-{z}_{2}\right)\right)}_{0}\right),{F}_{T0}\right)$$

30

Applying (7), (9), (11)

$${M}_{1}\cdot {\ddot{z}}_{1}\left(t\right)=\frac{{M}_{1}}{{M}_{1}+{M}_{2}}\cdot \left({F}_{1}\left(t\right)+{F}_{2}\left(t\right)\right)-K\cdot luz\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right),{\left({z}_{1}-{z}_{2}\right)}_{0}+\frac{{F}_{T0}}{K}\right)$$

31

$${M}_{2}\cdot {\ddot{z}}_{2}\left(t\right)=\frac{{M}_{2}}{{M}_{1}+{M}_{2}}\cdot \left({F}_{1}\left(t\right)+{F}_{2}\left(t\right)\right)+K\cdot luz\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right),{\left({z}_{1}-{z}_{2}\right)}_{0}+\frac{{F}_{T0}}{K}\right)$$

32

**Note 3**

When external forces are zeroes

$${M}_{1}\cdot {\ddot{z}}_{1}\left(t\right)=\left\{\begin{array}{ccc}-C\cdot tar\left(\left({\dot{z}}_{1}\left(t\right)-{\dot{z}}_{2}\left(t\right)\right),\frac{{F}_{T0}}{C}\right)-K\cdot luz\left(\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right)\right),{\left({z}_{1}-{z}_{2}\right)}_{0}\right)& if& {\dot{z}}_{1}\left(t\right)\ne {\dot{z}}_{2}\left(t\right)\\ luz\left(-K\cdot luz\left({\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right),\left({z}_{1}-{z}_{2}\right)\right)}_{0}\right),{F}_{T0}\right)& if& {\dot{z}}_{1}\left(t\right)={\dot{z}}_{2}\left(t\right)\end{array}\right.$$

33

$${M}_{2}\cdot {\ddot{z}}_{2}\left(t\right)=\left\{\begin{array}{ccc}C\cdot tar\left(\left({\dot{z}}_{1}\left(t\right)-{\dot{z}}_{2}\left(t\right)\right),\frac{{F}_{T0}}{C}\right)+K\cdot luz\left(\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right)\right),{\left({z}_{1}-{z}_{2}\right)}_{0}\right)& if& {\dot{z}}_{1}\left(t\right)\ne {\dot{z}}_{2}\left(t\right)\\ -luz\left(-K\cdot luz\left({\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right),\left({z}_{1}-{z}_{2}\right)\right)}_{0}\right),{F}_{T0}\right)& if& {\dot{z}}_{1}\left(t\right)={\dot{z}}_{2}\left(t\right)\end{array}\right.$$

34

Using (7), (9), (11) we obtain

$${M}_{1}\cdot {\ddot{z}}_{1}\left(t\right)=\left\{\begin{array}{ccc}-C\cdot tar\left(\left({\dot{z}}_{1}\left(t\right)-{\dot{z}}_{2}\left(t\right)\right),\frac{{F}_{T0}}{C}\right)-K\cdot luz\left(\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right)\right),{\left({z}_{1}-{z}_{2}\right)}_{0}\right)& if& {\dot{z}}_{1}\left(t\right)\ne {\dot{z}}_{2}\left(t\right)\\ -K\cdot luz\left(\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right)\right),{\left({z}_{1}-{z}_{2}\right)}_{0}+\frac{{F}_{T0}}{K}\right)& if& {\dot{z}}_{1}\left(t\right)={\dot{z}}_{2}\left(t\right)\end{array}\right.$$

35

$${M}_{2}\cdot {\ddot{z}}_{2}\left(t\right)=\left\{\begin{array}{ccc}C\cdot tar\left(\left({\dot{z}}_{1}\left(t\right)-{\dot{z}}_{2}\left(t\right)\right),\frac{{F}_{T0}}{C}\right)+K\cdot luz\left(\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right)\right),{\left({z}_{1}-{z}_{2}\right)}_{0}\right)& if& {\dot{z}}_{1}\left(t\right)\ne {\dot{z}}_{2}\left(t\right)\\ K\cdot luz\left(\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right)\right),{\left({z}_{1}-{z}_{2}\right)}_{0}+\frac{{F}_{T0}}{K}\right)& if& {\dot{z}}_{1}\left(t\right)={\dot{z}}_{2}\left(t\right)\end{array}\right.$$

36

In this case the lack of movement \({\dot{z}}_{1}\left(t\right)={\dot{z}}_{2}\left(t\right)\) occurs when \(\left|{z}_{1}\left(t\right)-{z}_{2}\left(t\right)\right|\le {\left({z}_{1}-{z}_{2}\right)}_{0}+\frac{{F}_{T0}}{K}\) (37)

The equations (25), (26) enable important parametrically made simplifications. This is presented below:

**Note 4**

When \({M}_{2}\to \infty\) the state \({\ddot{z}}_{2}\left(t\right)=0\) must be steady. It means a blockade of this block. So also \({\dot{z}}_{2}\left(t\right)=0\) and \({z}_{2}\left(t\right)=0\). The model passes to the single-mass system model (for the mass M1).

$${M}_{1}\cdot {\ddot{z}}_{1}\left(t\right)=\left\{\begin{array}{ccc}-C\cdot tar\left(\left({\dot{z}}_{1}\left(t\right)\right),\frac{{F}_{T0}}{C}\right)-K\cdot luz\left(\left({z}_{1}\left(t\right)\right),{\left({z}_{1}-{z}_{2}\right)}_{0}\right)+{F}_{1}\left(t\right)& if& {\dot{z}}_{1}\left(t\right)\ne 0\\ \frac{{M}_{1}/{M}_{2}}{{M}_{1}/{M}_{2}+1}\cdot \left({F}_{1}\left(t\right)+{F}_{2}\left(t\right)\right)+luz\left(-K\cdot luz\left({\left({z}_{1}\left(t\right),\left({z}_{1}-{z}_{2}\right)\right)}_{0}\right)+\frac{{F}_{1}\left(t\right)-{M}_{1}/{M}_{2}\cdot {F}_{2}\left(t\right)}{{M}_{1}/{M}_{2}+{1}_{2}},{F}_{T0}\right)& if& {\dot{z}}_{1}\left(t\right)=0\end{array}\right.$$

38

And finally

$${M}_{1}\cdot {\ddot{z}}_{1}\left(t\right)=\left\{\begin{array}{ccc}-C\cdot tar\left(\left({\dot{z}}_{1}\left(t\right)\right),\frac{{F}_{T0}}{C}\right)-K\cdot luz\left(\left({z}_{1}\left(t\right)\right),{\left({z}_{1}-{z}_{2}\right)}_{0}\right)+{F}_{1}\left(t\right)& if& {\dot{z}}_{1}\left(t\right)\ne 0\\ luz\left(-K\cdot luz\left({\left({z}_{1}\left(t\right),\left({z}_{1}-{z}_{2}\right)\right)}_{0}\right)+{F}_{1}\left(t\right),{F}_{T0}\right)& if& {\dot{z}}_{1}\left(t\right)=0\end{array}\right.$$

39

When external excitation *F**1**(t)* is absent

$${M}_{1}\cdot {\ddot{z}}_{1}\left(t\right)=\left\{\begin{array}{ccc}-C\cdot tar\left(\left({\dot{z}}_{1}\left(t\right)\right),\frac{{F}_{T0}}{C}\right)-K\cdot luz\left(\left({z}_{1}\left(t\right)\right),{\left({z}_{1}-{z}_{2}\right)}_{0}\right)& if& {\dot{z}}_{1}\left(t\right)\ne 0\\ -K\cdot luz\left({z}_{1}\left(t\right),{\left({z}_{1}-{z}_{2}\right)}_{0}+\frac{{F}_{T0}}{K}\right)& if& {\dot{z}}_{1}\left(t\right)=0\end{array}\right.$$

40

This model is useful for description full dynamics of the mass M1 interacting with the mass M2 when M2 > > M1.

**Note 5**

When \({M}_{1}\to 0\) the state \({\ddot{z}}_{2}\left(t\right)=0\) must be steady. the movement of massless element has a kinetic character. Firstly we analyze the Eq. (25) for the zeroed mass M1. Now:

$$0=\left\{\begin{array}{ccc}-C\cdot tar\left(\left({\dot{z}}_{1}\left(t\right)-{\dot{z}}_{2}\left(t\right)\right),\frac{{F}_{T0}}{C}\right)-K\cdot luz\left(\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right)\right),{\left({z}_{1}-{z}_{2}\right)}_{0}\right)+{F}_{1}\left(t\right)& if& {\dot{z}}_{1}\left(t\right)\ne {\dot{z}}_{2}\left(t\right)\\ luz\left(-K\cdot luz\left({\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right),\left({z}_{1}-{z}_{2}\right)\right)}_{0}\right)+{F}_{1}\left(t\right),{F}_{T0}\right)& if& {\dot{z}}_{1}\left(t\right)={\dot{z}}_{2}\left(t\right)\end{array}\right.$$

41

For \({\dot{z}}_{1}\left(t\right)\ne {\dot{z}}_{2}\left(t\right)\), using (6):

$${\dot{z}}_{1}\left(t\right)-{\dot{z}}_{2}\left(t\right)=\frac{1}{C}\cdot luz\left(-K\cdot luz\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right),{\left({z}_{1}-{z}_{2}\right)}_{0}\right)+{F}_{1}\left(t\right),{F}_{T012}\right)$$

42

For \({\dot{z}}_{1}\left(t\right)={\dot{z}}_{2}\left(t\right),\) according to (41), \(0=luz\left(-K\cdot luz\left({\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right),\left({z}_{1}-{z}_{2}\right)\right)}_{0}\right)+{F}_{1}\left(t\right),{F}_{T0}\right)\) (43)

So the massless element movement is described by equation:

\({\dot{z}}_{1}\left(t\right)={\dot{z}}_{2}\left(t\right)+\frac{1}{C}\cdot luz\left(-K\cdot luz\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right),{\left({z}_{1}-{z}_{2}\right)}_{0}\right)+{F}_{1}\left(t\right),{F}_{T012}\right)\) for any \({\dot{z}}_{1}\left(t\right),{\dot{z}}_{2}\left(t\right)\) (44)

Now, we analyze the Eq. (26) for the mass M2. By summation (41) and (25) we obtain:

$${M}_{2}\cdot {\ddot{z}}_{2}\left(t\right)=\left\{\begin{array}{ccc}{F}_{1}\left(t\right)+{F}_{2}\left(t\right)& if& {\dot{z}}_{1}\left(t\right)\ne {\dot{z}}_{2}\left(t\right)\\ {F}_{1}\left(t\right)+{F}_{2}\left(t\right)& if& {\dot{z}}_{1}\left(t\right)={\dot{z}}_{2}\left(t\right)\end{array}\right.$$

45

This means that

\({M}_{2}\cdot {\ddot{z}}_{2}\left(t\right)={F}_{1}\left(t\right)+{F}_{2}\left(t\right)\) for any \({\dot{z}}_{1}\left(t\right),{\dot{z}}_{2}\left(t\right)\) (46)

Final form of the model for any \({\dot{z}}_{1}\left(t\right),{\dot{z}}_{2}\left(t\right)\) is done by equations (44) and (46).

When also

\(-{F}_{T0}\le -K\cdot luz\left({\left({z}_{1}\left(t\right)-{z}_{2}\left(t\right),\left({z}_{1}-{z}_{2}\right)\right)}_{0}\right)+{F}_{1}\left(t\right)\le {F}_{T0}\) we have \({\dot{z}}_{1}\left(t\right)-{\dot{z}}_{2}\left(t\right)=0\) (47)

It means the stiction state between the blocks. It is independent of the second mass action.

**Note 6**

When \({M}_{2}\to \infty\) and \({M}_{1}\to 0\) (see the notes 4 and 5) the mass M2 is treated as immobile and the movement of massless element has a kinetic character. Here the model has a form:

$${\dot{z}}_{1}\left(t\right)=\frac{1}{C}\cdot luz\left(-K\cdot luz\left({z}_{1}\left(t\right),{\left({z}_{1}-{z}_{2}\right)}_{0}\right)+{F}_{1}\left(t\right),{F}_{T012}\right)$$

48

When also \({F}_{1}\left(t\right)=0\), taking into account (7), (9) and (11):

$${\dot{z}}_{1}\left(t\right)=-\frac{K}{C}\cdot luz\left(\left({z}_{1}\left(t\right),{\left({z}_{1}-{z}_{2}\right)}_{0}\right)+\frac{{F}_{T012}}{K}\right)$$

49

In this case we can implement a new “freeplay-friction” parameter: \(ff={\left({z}_{1}-{z}_{2}\right)}_{0}+\frac{{F}_{T012}}{K}\) (50)

**Note 7**

The presented models have no entanglements. This statement is very important for simulation calculations!

**Note 8**

The presented models can be easy reformulated to the models basing on luz(…) and tar(,,,) when kinetic dry friction characteristics have more sophisticated forms (e.g. with taking into account the Stribeck effect), and when parameters of kinetic and static dry friction forces are not the same.

**Note 9**

The presented models can be easy reformulated to the models describing rotative systems, especially systems with gear elements (example system in Fig. 5). In these cases the masses must be replaced by moments of inertia, etc. This method can be implemented also for a synthesis of the model of vehicle steering system with inclusion freeplay and friction elements.

Several ready-to-use models of mechanical systems operating with freeplay and/or friction action (also when there is a problem of static indeterminacy) have been used and referred in publications [26, 34, 38, 43, 44]. Of course the models of multibody systems with many freeplay / friction actions can have very complicated forms, but fortunately in many cases without variable entanglement.