where:


In order to evaluate the last remaining term *Q*ϕ to be used in the (A1), let us consider the elementary work of all active forces related to an infinitesimal rotation δϕ:

$$
\delta \delta \mathcal{W}\_{\varphi} = -\mathbf{N} \cdot \mathbf{B} \delta \delta \varphi + f \mathbf{N} \cdot \mathbf{T} \mathbf{O} \delta \varphi + \mathbf{R}\_{\hat{\mathbf{k}}} \cdot \mathbf{E} \mathbf{N} \cos \varphi \, \delta \varphi + \mathbf{G} \cdot \mathbf{l} \cos \varphi \, \delta \varphi - \mathbf{S}\_{T} \cdot \mathbf{O} \mathbf{O}^{\prime} \delta \varphi \tag{A14}$$

where:


The generalized force *Q*ϕ acting on the dynamic system is, thus, the algebraic sum of the moments of all active forces relative to point *O*:

$$Q\_{\overline{\rho}} = \frac{\delta W\_{\overline{\rho}}}{\delta \overline{\rho}} = -\mathbf{N} \cdot \mathbf{B} \mathbf{S} + f \mathbf{N} \cdot \mathbf{T} \mathbf{O} + \mathbf{R}\_{\overline{\mathbf{k}}} \cdot \mathbf{EN} \cos \overline{\rho} + \mathbf{G} \cdot \mathbf{l} \cos \overline{\rho} - \mathbf{S} \mathbf{r} \cdot \mathbf{O} \mathbf{O}^{\prime} \tag{\text{A15}}$$


• remembering that *OD* [m] is the height of the frame of the machine;


• *OO* = *OD*·*cos*(<sup>θ</sup> + ϕ)

and, so, Equation (A15) can be written as:

$$\begin{split} Q\_{\varphi} &= -\mathbf{N} \cdot \mathbf{L} \cdot \cos(\alpha + \varphi) + f \mathbf{N} \cdot [r + L \cdot \sin(\alpha + \varphi)] + \\ &+ R\_{k} \cdot r\_{1} \cos \varphi + G \cdot l \cos \varphi - S\_{T^{\ast}} [\mathbf{O} \mathbf{D} \cdot \cos(\theta + \varphi)] \end{split} \tag{A16}$$

Is it now possible, using (A11), (A12) and (A15) to write four of the five terms that appear in (A1).

$$\frac{\partial T}{\partial \dot{q}} = \left(l\_{\theta \mathcal{Y}} + m \cdot L^2\right) \dot{q} \implies \frac{d}{dt} \Big(\frac{\partial T}{\partial \dot{q}}\Big) = \left(l\_{\theta \mathcal{Y}} + m \cdot L^2\right) \ddot{q} \tag{A17}$$

$$\frac{\partial T}{\partial p} = 0 \tag{A18}$$

$$\frac{\partial P}{\partial \boldsymbol{\varphi}} = 2\boldsymbol{c} \cdot \boldsymbol{L}^2 \cdot \boldsymbol{\varphi} \tag{A19}$$

$$\frac{\partial \mathcal{R}}{\partial \dot{\phi}} = 2\mu \cdot L^2 \cdot \dot{\phi} \tag{A20}$$

Finally, using (A17), (A18), (A16), (A19), and (A20) it is possible to write the final expression of (A1):

$$\begin{aligned} & \left( \mathrm{I}\_{0y} + mL^2 \right) \ddot{\varphi} + 2c\mathrm{I}^2 \varphi + 2\mu \mathrm{I}^2 \dot{\varphi} = \\ &= -NL\cos(a+\varphi) + fN[r+L\sin(a+\varphi)] + R\_k r\_1 \cos\varphi + Gl \cos\varphi - \mathrm{S}\_T [\mathrm{OD}\cos(\theta+\varphi)] \end{aligned} \tag{A21}$$

that can be written in the following form:

$$\begin{aligned} \ddot{\boldsymbol{\varrho}} + \frac{2L^2}{l\_{\text{sy}} + mL^2} (\text{cpt} + \mu \dot{\boldsymbol{\phi}}) &= \\ \frac{1}{L\_{\text{sy}} + mL^2} \left\{ -NL \cos(\alpha + \phi) + f \mathbf{N} [\boldsymbol{r} + L \sin(\alpha + \phi)] + R\_k \mathbf{r}\_1 \cos \boldsymbol{\varrho} + \mathbf{G} \boldsymbol{l} \cos \boldsymbol{\varrho} - S\_{\text{I}} \boldsymbol{\omicron} \boldsymbol{\omicron} \cos(\theta + \phi) \; \; \; \; \theta \boldsymbol{\varrho} \right\} \end{aligned} \tag{A22}$$

#### **Appendix B**

The modulus of the normal reaction *N* between the feeler wheels and the soil can be determined from the equilibrium condition of the system at a fixed time, setting equal to zero the algebraic sum of the moments of all forces acting on the system relative to point *O*.

The above-mentioned condition is raised, when (A16):

$$\begin{cases} Q\_{\varphi} = -\text{NL}\cos(a+\varphi) + f\text{N}[r+L\sin(a+\varphi)] + \\ + R\_{k}r\_{1}\cos\varphi + Gl\cos\varphi - S\_{T}[OD\cos(\theta+\varphi)] = 0 \end{cases} \tag{A23}$$

As the angles α and ϕ in (A23) are time-dependent, it is convenient to fix the time at which the point of contact between the feeler wheels and the profile of soil roughness is in the upper most part of the profile itself: In this case (Figure 2) α = ϕ = 0 so (A23) simplifies in:

$$Q\_{\varphi} = -NL + fNr + R\_kr\_1 + Gl - S\_T OD \cos \theta = 0 \tag{A24}$$

and, finally:

$$N = \frac{R\_k r\_1 + Gl - S\_T OD \cos \theta}{L - fr} \tag{A25}$$
