# Experimental Verification of the Dynamic Model of Turbine Blades Coupled by a Sealing Strip
## 4 The Numerical Model  
Figure 8 shows a mind map of this paper intentions. The experimental set-up described in Section 2 is replicated using a state-of-the art numerical tool first described in [29]. The ultimate purpose is that of gaining a deeper understanding of the strip working conditions through the ensuing experimental-numerical comparison.  
This section explores the \"building blocks\" of the numerical tool, necessary to perform the nonlinear simulation.  
1. **Finite Element Models**: the blades and the strip are modeled using FE (Finite Elements). In the present case, Ansys Mechanical APDL 16.1 is chosen. No contact is imposed between blades and strip at this stage, therefore the two blades and the strip are handled into two separate files. Nevertheless, the two separate models should have the same reference system and a coincident regular mesh on the contact surfaces.  
Figure 8: Functional scheme of the numerical simulation and its links to the experimental activity.  
Figure 7: (**a**) IP and (**b**) OOP FRFs for different excitation levels, tests repeated after having switched off and on the compressed air system.  
2. **Boundary Conditions**: the nodes on the base of the blades are constrained (zero displacement along all DOFs) to reproduce the clamping imposed by the hydraulic machine.
3. **CB-CMS reduction**: The size of the two models is reduced by performing a Craig-Bampton Component Mode Synthesis (CB-CMS) reduction [33; 34]. This reduction technique requires the selection of master (i.e., physical) and slave (i.e., modal) degrees of freedom. The physical DOFs include: * two nodes, one for each blade, to apply the shaker excitation force. The nodes are chosen on the outer part of the blade platforms, to replicate the experimental set-up; * two nodes at the tip of the two blades, in the same position where the accelerometers are placed. These nodes are used to record the blades' response; * a total of 200 contact nodes (50 for each of the four contact interfaces on blades and strip). As pointed out in [29], whenever a flexible structure is involved, ensuring an adequate representation of the contact requires a high number of contact nodes.
4. **Nonlinear simulation**. The two reduced models are imported in a Matlab environment (R2016a version, Mathworks, Natick, MA, USA) where a state-of-the-art code is used to compute the nonlinear forced response of the system. A few highlights of the code, closely analyzed in [29] can be found below. * The multi-Harmonic Balance method is used to solve the nonlinear equilibrium equations in the frequency domain: \[(-\omega^{2}M+i\omega C+K)\cdot x=F_{e}+F_{nl},\] (1) where the mass and stiffness matrices \(M\) and \(K\) of the blade strip system come from the CB-CMS reduction. The structural damping of the two bodies, i.e., matrix \(C\) is assumed to be modal. The value of modal damping \(\zeta\) is tuned using the Free curves (i.e., no strip) and here set at 0.0002. The vector \(F_{e}\) contains the two shaker forces, whose amplitude \(F_{E}\) and phase is set equal to those investigated in Section 3.2. The vector \(F_{nl}\) contains the nonlinear contact forces. * At this stage blades and strip should be regarded as a unique system. The node-to-node contact between blades platforms and strip is ensured by a state of the art contact model implementing a normal and a tangential contact spring per node (see also Figure 8). The model is capable of simulating the three fundamental contact states: stick (the nodes are connected by the two springs), slip (the normal spring is still present while the tangential direction moves freely) and lift-off (no contact). The contact parameters' values influence the result as further commented on in Section 5.2. * The Jacobian Alert Algorithm, first presented in [35], ensures a fast yet accurate calculation.