# Viscoelastic Dampers for Rotors: Modeling and Validation at Component and System Level
## 2 Viscoelastic Model Identification
### Reference Model for Parametric Identification  
Ideal elastic materials do not dissipate energy when deformed and can be represented with only a linear spring (a in Figure 4). However, since no actual material is ideal elastic, the simplest way to model material damping is to assume that the material reacts with a force depending only on the strain rate. These two basic behaviors can be combined to obtain a more accurate representation of the material exploiting a simple series of a spring and a damper (b) or by means of more elaborate spring-damper models like [PERSON]'s (c) and Wiechert's (d) representation. Figure 3: Cross-section of the machine from lateral view. (1) Mobile part of the actuator. (2) Collet for actuator support. (3) Static part of the actuator. (4) Screw to fix actuator to the machine. (5) Plexiglass cover. (6) Top nut. (7) Metallic spacer. (8) Metallic cover of the actuator. (9) Body of the machine. (10) Seismic mass. (11) External support of the damper. (12) Internal support of the damper tower. (13) Viscoelastic damper. (14) Displacement sensor. These models represent the equilibrium restoring stiffness with an isolated spring \(K_{E}\) in parallel with one (c) or several (d) [PERSON]'s arms. In the case of configuration (c), the force provided by the elastomeric material is obtained as the sum of the force in the Maxwell arm and that in the isolated spring  
\[F=F_{M}+F_{E} \tag{1}\]  
The force in the Maxwell arm in Laplace domain is depending on the traction/compression \(x\) applied to the material and is equal to:  
\[F_{M}\ =\ \frac{k_{1}s}{s+\frac{k_{1}}{C_{1}}}x \tag{2}\]  
Adding the force \(F_{E}=k_{E}x\) in the equilibrium spring, the total force is  
\[F\ =\ k_{E}x\ +\ \frac{k_{1}s}{s+\frac{k_{1}}{C_{1}}}x\ =\ \left\{k_{E}+\frac{k_{1}s}{s+\frac{k_{1}}{C_{1}}} \right\}x \tag{3}\]  
This modeling approach can be extended to the case of [PERSON]'s representation (d), where the total force \(F\) is the sum of the components of each parallel arm  
\[F=F_{E}+\sum_{j}F_{j} \tag{4}\]  
Extending from Equation (2),  
\[F_{j}=\frac{k_{j}s}{s+\frac{k_{j}}{c_{j}}}x \tag{5}\]  
and  
\[F\ =\ F_{E}\ +\ \sum_{j}F_{j}\ =\ \left\{k_{E}+\sum_{j}\frac{k_{j}s}{\left(s+ \frac{k_{j}}{c_{j}}\right)}\right\}x \tag{6}\]  
follows. In this paper, [PERSON]'s model is adopted to represent the behavior of the dampers under test effectively. However, the proposed methodology is valid also with [PERSON]'s representation should it be necessary to take relaxation time distribution into account. Figure 4: Most common representations of viscoelastic material. (**a**) Ideal elastic material ([PERSON]’s representation); (**b**) Combination of restoring and damping force ([PERSON]’s representation); (**c**) [PERSON]’s representation; and, (**d**) [PERSON]’s representation. The characterization of the viscoelastic dampers under test is based on the following representation of the system  
\[H\ =\ \frac{F_{ACT}}{x}\ =\ ms^{2}\ +\ \left(k_{E}+\frac{k_{1}s}{s+\frac{k_{1}}{C_{1 }}}\right)\ +\ \frac{K_{ACT}s}{s+\frac{K_{ACT}}{C_{ACT}}}, \tag{7}\]  
where \(H\) is the transfer function between the traction/compression \(x\) of the elastomer, and the force \(F_{ACT}\) applied on it, \(m\) is the inertial action obtained as the sum of the mass of the damper support ((11) in Figure 3) and of the mobile part of the actuator (1). The second component in Equation (7) is the representation of the damper, while the third part models the losses on the actuator due to eddy currents generation, which is represented by a dynamic identified with FEM (Finite Element Method) computation. This has been performed with the software COMSOL Multiphysics 4.3a (Comsol Inc., Stockolm, Sweden, May 2012) following the indications reported in [10], where it is demonstrated that the mechanical impedance of an eddy current damper can be represented as a series of a viscous damper and a spring.