## 3 Statistical Tolerance Analysis of 3D Printed Non-Assembly Mechanisms in Motion
### Tolerance Analysis of Systems in Motion Considering Joint Clearance  
As mentioned in Section 2.1, there are different approaches for the tolerance analysis of systems in motion. In this contribution, the vector loop approach is used due to its suitability to systems in motion and the possibility of considering joint clearance according  
Figure 2: Framework for the statistical tolerance analysis of 3D-printed non-assembly mechanisms using empirical predictive models. to [11]. To show the general idea of the vector loop approach for planar non-assembly mechanisms, it is applied in the following to a schematic 4-bar mechanism with two kinematic joints affected by clearance (\(J_{12}\) and \(J_{23}\)), whereby each of the linkages \(L_{j}\) is thus represented as a vector (see Figure 3). The clearance is hereby modelled as the virtual, massless clearance vector \(\vec{c}_{ij}\) connecting the two centre points of the joint pair as can be seen in Figure 3c [33]. For determining the degrees of freedom (DoF) and thus the required number of vector loop equations \(p\) for characterising the motion behaviour, the equation according to [PERSON] for a mechanism consisting of \(g\) linkages and \(n\) joints is used [34]:  
\[p=g-(n-1). \tag{1}\]  
Applying this equation to the shown mechanism including four linkages \(L_{j}\) and four joints \(J_{ij}\), it becomes evident that one vector loop is sufficient for the characterisation of its motion behaviour. The resulting equation consists of a real and an imaginary part, which both must be equal to zero. In the following, only the summarised form of the equation will be shown (for further information, please refer to [11]):  
\[L_{1}\cdot e^{j\theta_{1}}+c_{12}\cdot e^{j\gamma_{12}}+L_{2}\cdot e^{j\theta _{2}}+c_{23}\cdot e^{j\gamma_{23}}-L_{AB}-L_{3}\cdot e^{j\theta_{3}} = 0. \tag{2}\]  
As for this non-linear vector loop equation, an explicit solution is difficult to find, a numerical solution is required. Therefore, a numerical and iterative method needs to be applied in order to determine the unknown angles \(\theta\) of each joint \(J_{ij}\) for every discretisation step in order to reproduce the motion behaviour of the planar mechanism. In this paper, the vector loop approach is illustratively applied to planar mechanisms, however it is also applicable for three-dimensional mechanisms as the model can be enhanced by adding another equation for the third dimension [6; 35]. The dimensional length deviation of the linkages \(L_{j}\) and their influence on the motion behaviour are thereby considered through the integration of empirical predictive models. For the consideration of the clearance vectors, the missing information such as their lengths \(c_{ij}\) and the force angle \(\gamma\) have to be known a priori for solving the equation. In order to determine the missing information for solving the vector loop equation, including joint clearance, a MBS is purposeful. With the help of MBS, systems in motion can be simulated using commercial software tools. Hereby the systems can be described by mathematical substitute models. Based on these dynamic models, joint forces and angles of kinematic joints can be determined and thus be used for solving the vector loop equation in the tolerance analysis for the nominal mechanism without deviations [33]. Assuming the surface of the joints is rigid and there is no friction, the direction of the clearance vector coincides with the normal direction of the collision plane. With this premise, the clearance vector \(\vec{c}\), consisting of the clearance \(c\) and the force angle \(\gamma_{ij}\), points in the same direction  
Figure 3: (a) Vector loop of a 4-bar-mechanism with two kinematic joints affected by clearance. (b) Schematic representation of joint clearance for kinematic joint \(J_{23}\) (joint clearance illustrated exaggerated). (c) Schematic representation of clearance vector \(\vec{c}_{23}\) for kinematic joint \(J_{23}\). as the joint force (see Figure 3c) [33].