# An Outline of Fused Deposition Modeling: System Models and Control Strategies
## 2 Systems
### Extrusion
#### 2.1.2 Filament Extruder  
In FFF, extrusion occurs by pushing the filament into a heated chamber called liquefier, in which the material is melted and directed to the nozzle (see Figure 2) [32]. The motion of the filament is induced by a couple of electrical motors that drive a roller each. A roller is a dented gear fixed to the end-effector whose ideal rotation impresses motion on the filament through friction in the form:  
\[v_{f}=\omega r, \tag{4}\]  
where \(v_{f}\) is the linear speed of the filament, \(\omega\) the rotation speed of the rollers and \(r\) the radius of the rollers at the gear teeth. Loss of ideality can arise in terms of slippage or deformation phenomena. * Slippage \(\lambda\) causes loss of speed given by \[v_{f}=\frac{\omega r}{1-\lambda},\] (5)  
Figure 2: Scheme of the extruder in FFF applications. Image from reference [33]. and can be induced by roundness deviation, wear or insufficient torque;
* Deformation occurs in the filament indentation as a result of the compression exerted by the roller teeth, which can cut into the filament. Slippage and deformation can lead to under-extrusion, and extreme slippage causes filament shredding that can clog the extruder [34]; it is therefore important that these phenomena are avoided. The rollers push the filament into the liquefier, where the phase shift occurs. The pressure exerted by the descending filament pushes the molten material through the nozzle. Based on the connection between the driving motors and the liquiefier, two types of extruder can be identified:
* Direct: the motors are mounted directly on top of the liquefier. The load on the end-effector is higher, but the movement of the filament is more constrained and hence more controlled;
* Bowden: only the liquefier is mounted on the printing head, while the motors are placed in a fixed position of the machine, diminishing the load on the end-effector. However, the filament needs to cover some distance from the rollers to the liquefier, which may lead to bending and buckling. The liquefier has been modeled analytically [35] both in the time and the frequency domains and the models obtained have been validated experimentally, reaching satisfactory prediction accuracy. The compression force \(\bar{F}\) to be applied to the filament to extrude the melt is derived as \[\bar{F}=A\Delta p,\] (6) where \(\Delta p\) is the total pressure drop along the liquefier, and \(A\) is the cross-section of the filament (which is equal to the cross section of the liquefier). By imposing that the two drive rollers apply a total force \(F\) on the filament that equals the required one \(\bar{F}\) \[F=\bar{F},\] (7) the corresponding torque \(T\) exerted by each of the two drive rollers under ideal conditions can be written as \[T=\frac{\bar{F}}{2}r.\] (8) A first-order transfer function \(G(s)\) (where \(s\) is the complex frequency) between the Laplace transform of the reference flow rate \(U(s)\) and the transformed measured flow rate \(Y(s)\) with time delay \(\tau\) that approximates the liquefier dynamics at lower frequencies has also been developed \[G(s)=\frac{Y(s)}{U(s)}=\frac{\mu}{T_{0}s+1}e^{-\tau s},\] \[T_{0}=\frac{L}{R},\] (9) \[\mu=\frac{1}{R}.\]  
where \(T_{0}\) is the time constant of the system, \(\mu\) the static gain, \(R\) models the slippage between rollers and filament and \(L\) represents the properties of the material and the heat flux. The main assumptions of this derivation are that (i) the material behaves as a generalized Newton fluid; (ii) the flow is isothermal at steady-state but not during transient states, in which the viscosity is corrected based on temperature; (iii) slippage is present; and (iv) the torque is upper bounded. Nevertheless, the flow rate is computed by approximating the extruder to a pipe, thus equaling the relationship between input and output flow  
\[q_{in}=q_{out}\rightarrow\frac{v_{f}}{\pi r^{2}}=\frac{v_{\varepsilon}}{\pi r _{\text{nozzle}}^{2}} \tag{10}\]according to the mass conservation principle.