## 2 Models and Methods
### Substructure Coupling  
Once the models of the component subsystems of the agricultural machine are defined, they can be assembled using the substructure coupling technique. For a mechanical system consisting of \(n\) linear subsystems, the equation of motion of each subsystem \(r\) may be expressed as:  
\[\mathbf{M}^{(r)}\ddot{\mathbf{u}}(t)^{(r)}+\mathbf{C}^{(r)}\dot{\mathbf{u}}(t)^{(r)}+ \mathbf{K}^{(r)}\mathbf{u}(t)^{(r)}=\mathbf{f}(t)^{(r)}+\mathbf{g}(t)^{(r)} \tag{1}\]  
where:  
\(\mathbf{M}^{(r)}\), \(\mathbf{C}^{(r)}\), \(\mathbf{K}^{(r)}\) are the matrices accounting for the mass, damping, and stiffness;  
\(\mathbf{u}(t)^{(r)}\) is the displacement vector;  
\(\mathbf{f}(t)^{(r)}\) is the external force vector;  
\(\mathbf{g}(t)^{(r)}\) is the internal constraint force vector, accounting for the connecting forces with other subsystems. The equations of motion of \(n\) subsystems can be gathered in a block diagonal format as:  
\[\mathbf{M}\ddot{\mathbf{u}}(t)+\mathbf{C}\dot{\mathbf{u}}(t)+\mathbf{K}\mathbf{u}(t)=\mathbf{ f}(t)+\mathbf{g}(t), \tag{2}\]  
with:  
\[\mathbf{M}= \begin{bmatrix}\mathbf{M}^{(1)}&&\\ &\ddots&\\ &&\mathbf{M}^{(n)}\end{bmatrix} ,\quad\mathbf{C}= \begin{bmatrix}\mathbf{C}^{(1) }&&\\ &\ddots&\\ &&\mathbf{C}^{(n)}\end{bmatrix} ,\quad\mathbf{K}= \begin{bmatrix}\mathbf{K}^{(1) }&&\\ &\ddots&\\ &&\mathbf{K}^{(n)}\end{bmatrix}  \tag{3}\]  
\[\mathbf{u}(t)= \begin{Bmatrix}\mathbf{u}(t)^{(1)}\\ \vdots\\ \mathbf{u}(t)^{(n)}\end{Bmatrix} ,\quad\mathbf{f}(t)= \begin{Bmatrix}\mathbf{f}(t)^{(1)}\\ \vdots\\ \mathbf{f}(t)^{(n)}\end{Bmatrix} ,\quad\mathbf{g}(t)= \begin{Bmatrix}\mathbf{g}(t)^{(1)}\\ \vdots\\ \mathbf{g}(t)^{(n)}\end{Bmatrix}  \tag{4}\]  
The compatibility condition is generally expressed as:  
\[\mathbf{B}\mathbf{u}(t)=\mathbf{0} \tag{5}\]  
where each row of \(\mathbf{B}\) states that the displacement of a pair of matching DoFs, i.e., DoF \(l\) belonging to subsystem \(r\) and DoF \(m\) belonging to subsystem \(s\), must be the same (\(u_{l}^{(r)}-u_{m}^{(s)}=0\)). The equilibrium condition for internal constraint forces is generally expressed as:  
\[\mathbf{L}^{T}\mathbf{g}(t)=\mathbf{0} \tag{6}\]  
Figure 4: Lumped parameter models of mounted or semi-mounted machines: (**a**) rear machine; (**b**) front machine. where each row of matrix \(\mathbf{L}^{T}\) states that: for any pair of matching DoFs, the sum of internal constraint forces must be zero (i.e., \(g_{l}^{(r)}+g_{m}^{(s)}=0\)); and for non connecting DoFs, the corresponding internal constraint forces must be zero. Note that the number of rows of the matrix \(\mathbf{L}^{T}\) is equal to the sum of the number of non connecting DoFs and the number of pairs of matching DoFs. Gathering Equations (2), (5) and (6), one obtains the three-field formulation that describes the coupling among \(n\) subsystems:  
\[\left\{ \begin{array}{l}\mathbf{M}\ddot{\boldsymbol{u}}(t)+\mathbf{C}\dot{ \boldsymbol{u}}(t)+\mathbf{K}\boldsymbol{u}(t)=\boldsymbol{f}(t)+\boldsymbol {g}(t)\\ \mathbf{B}\boldsymbol{u}(t)=\boldsymbol{0}\\ \mathbf{L}^{T}\boldsymbol{g}(t)=\boldsymbol{0}\end{array} \right. \tag{7}\]  
To solve the coupling problem, here a unique set of DoFs \(\boldsymbol{q}\) is considered, composed by the unique set of interface DoFs and by non-interface DoFs. By stating that  
\[\boldsymbol{u}(t)=\mathbf{L}\boldsymbol{q}(t), \tag{8}\]  
the interface equilibrium is automatically satisfied and the interface forces are eliminated from the first line of Equation (7). In fact, since the full set of DoFs \(\boldsymbol{u}\) is obtained from the unique set \(\boldsymbol{q}\), the compatibility condition becomes:  
\[\mathbf{B}\boldsymbol{u}(t)=\mathbf{BL}\boldsymbol{q}(t)=\boldsymbol{0} \tag{9}\]  
and holds for all values of \(\boldsymbol{q}\). Therefore, the compatibility condition, i.e., the second line in Equation (7), is always satisfied and the three-field formulation reduces to:  
\[\left\{