# A Method to Minimize the Effort for Damper-Blade Matching Demonstrated on Two Blade Sizes
###### Abstract
The Frequency Response Function (FRF) of the blade is the object of experimental observations that are compared with those produced by numerical models: this is the case from the early pioneering 1980 studies [1] through the decades down to the latest achievements where numerical models have reached a high level of complexity and, as increasingly claimed, \"fidelity\": see [6; 7; 8; 9; 10] just to quote a few. To achieve this goal, several numerical tools are necessary, e.g., a proper representation of friction contacts [11; 12; 13; 14], an efficient computational strategy to solve the nonlinear equilibrium equations in the frequency domain [15; 16; 17; 18] and finally methods to reduce the size of the nonlinear system of equations [4; 19; 20]. Apart from testing their predictive capabilities against experimental evidence, numerical codes can be used to perform optimization studies and sensitivity analyses during the design stage [7]. With this goal in mind, the authors have developed a 3-step optimization procedure. Step 1 is to filter out all those geometries that would lead to undesirable kinematics and contact forces [21; 22]. Step 2 is to identify those ranges of parameters that are better in terms of added stiffness and damping [23]. Step 3 comes from the need to complete the picture by exploring how the basic geometrical design parameters of blades interact with the damper parameters, then for which parameters combination a designer may find the best blade-damper match in view of the vibratory response under resonant excitation. While the first two steps are stand-alone, i.e., they are performed independently of the blades and only as a function of the damper shape and the relative displacement between adjacent blade platforms, the search for the best match between a damper and a given blade is very demanding from the computational point of view as it requires repeated dynamic nonlinear coupled analyses of the blade-damper system. The approach to Step 3 proposed in this paper aims at drastically reducing the size of the nonlinear system by adopting a special representation of the dynamic behavior of the blade. In detail, platform and neck can be accurately represented using simple beam elements, while realistic airfoils can be modeled using standard FE codes and Component Mode Synthesis (CMS) techniques. The advantages of this approach are:  
1. representing the kinematics of the platform by means of the displacements and the rotations of just one appropriate single point allows all the nonlinear contact forces distributed on the underplatform/damper interfaces to be represented by means of the \"dynamically equivalent\" concentrated forces and moments applied at that point;
2. modeling the neck by means of beams represents the reality in its most simple, straightforward and time-honored way, and is fully compatible with the advantages of describing the kinematics of cross-sections (of neck and platform) in terms of a single point displacements and rotations;
3. the airfoil CMS reduction is performed only once, as the airfoil is kept constant during optimization calculations. A consequence of point 1 is that, as shown in [24; 25], the whole effect of friction forces on a platform can be reduced to time-variable nonlinear forces and moments (half of them dynamically irrelevant due to neck shape), i.e., the size of the linear system to be solved reduces to that of one beam element node, independently of the number of contact points. This makes the proposed approach very convenient in capturing, representing, and clarifying the role of frictional phenomena on the blade/damper system. Section 2 of this paper offers a brief description of the method, i.e., the reduction strategy here called Platform Centered Reduction first presented in [25], proves its accuracy and shows a sample of results to demonstrate the potential of the platform-centered representation of friction forces. The novel contributions of the present paper can be found in Sections 3-5 and are summarized in the following.