Modeling and Testing Friction Flexible Dampers: Challenges and Peculiarities
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Abstract
This paper deals with the dynamic of blades with strip dampers. The purpose is 1) to present the results of the dynamic numerical calculation, 2) to demonstrate the need for the experimental data on the bladestrip contact to be used as input to the calculation, 3) to propose a new test rig design to obtain them and 4) to test the key components of the new test rig. The forced responses of two blades coupled by a strip damper are calculated at different excitation and centrifugal force values. The dependence of the numerical results on the contact parameter values is confirmed in this significant reference case. The design of a new test rig is then proposed: both the blade frequency response function and the contact hysteresis cycles at the bladestrip contact are measured. It is shown how contact parameters can then be derived from experimental data. The main novelty of the test rig here proposed is the strip loading system, which simulates the uniform pressure distribution provided by the centrifugal force in real operating conditions. This loading system is noncontact and uses compressed air. Classical loading systems which see dead weights directly connected to the strip are assessed and their expected inadequacy is confirmed. The compressed air system is tested by measuring the pressure produced between strip and blades: pressure is uniform across the contact patch, constant in time and its mean value corresponds to realistic pressure values actually experienced by strip dampers during service.
Keywords
Strip dampers Numerical modeling Test rig Contact parameters Forced responseIntroduction
Due to the high modal density of realistic bladed disks and to the broad frequency content of the aerodynamic excitation forces, attaining a blade design which is resonancefree in the frequency range of interest is unfeasible. Since turbine blades do not benefit significantly from material hysteresis and aerodynamic damping, the current best option is to add external sources of damping, e.g. in the form of dry friction devices [1, 2]. Dry friction can be incorporated into the blade design, in the form of shrouds, lacing wires or zigzag pins. Alternatively, external devices such as solid dampers (available in several designs, cylindrical, curved flat, wedge damper, etc), thinstrip dampers [3, 4] or ring dampers [5, 6] can be added to minimize the resonant blade response. A detailed analysis of the different sources of damping in turbo engines was performed in [2]. It was shown that the values of Q (amplitude factor, inverse of the loss factor \(\eta \)) may vary as follows: 3000 to 10,000 for material damping, 50 to 140 for root damping, 180 to 2500 for shroud damping, and 15 to 250 for platform damping (external dry friction dampers). Dampers are especially effective (low values of Q) if bending modes are considered (also explored in this paper), while Q values tend to increase for higher (torsional) modes. For this reason, the focus of the present investigation is on lower (bending) modes.
External dry friction dampers are then extensively used in turbine designs because they are not only very effective, but also easy to manufacture, install and substitute, relatively inexpensive and can withstand high temperatures. Among external dry friction dampers, the strip (or seal) dampers are thin flexible metallic strips which are positioned under the blade platforms of turbine bladed disks and their primary function is to seal the cooling air. They are pushed against the blade platforms by the centrifugal force. When relative motion takes place between strip and blade platforms, the friction forces dissipate vibrational energy and consequently the system response is damped. In this way the strips act also as friction dampers which introduce both damping and a constraint affecting significantly the blade natural frequencies [7].
The design of the strips, not only for sealing but also for damping purpose, represents a topic of recent interest for the turbine designers, as very few studies can be found in literature [3, 4].
In the past 15 years, the so called solid underplatform dampers (UPDs) have been extensively investigated [8, 9, 10, 11, 12, 13, 14]. Most of the proposed calculation methodologies for bulk UPDs use simplifying assumptions, such as neglecting the damper inertia or flexibility [14, 15, 16, 17, 18]. However, these assumptions, perfectly adequate for solid UPDs, cannot be applied to model strip dampers, as their high flexibility needs to be taken into account in the modeling process [4, 7]. Therefore, in this paper the strip damper is modeled using Finite Elements (FE) [4]. Furthermore, the most advanced stateoftheart calculation methods are here applied to predict the dynamics of two blades and a strip damper between them. In detail, the nonlinear forced response is calculated with high computational efficiency in the frequency domain using the MultiHarmonic Balance Method (MHBM) [19, 20], a reduction method is adopted to decrease the size of the blades [21] and the nonlinear equations are solved by an iterative solver using the analytical computation of the Jacobian matrix [20, 22]. Lastly, a novel technique presented in [22] ensures that the MHBM (an approximate method) offers results with an accuracy above a userdefined threshold.
Even if the calculated responses vary with the centrifugal to excitation force ratio following the expected pattern, it is shown that the computational results in terms of natural frequency and vibration amplitude strongly depend on the contact parameters values (friction coefficient, normal and tangential contact stiffnesses). This dependence has not been addressed in [3, 4] where a unique assumption on the contact parameter values has been used throughout the papers. This practice has potentially dangerous consequences as will be shown in a dedicated section of this paper. The contact parameters should be determined experimentally. However no record of this investigation performed on very flexible dampers can be found in literature. Furthermore, the results of similar investigation performed on solid UPDs [12, 13, 14, 23, 24, 25, 26, 27] cannot be applied here as normal loads and contact conditions of solid and flexible dampers differ.
In the past ten years, the AERMEC lab at the Polytechnic of Turin has devoted time and effort in the direct experimental investigation of friction parameters on different type of friction contacts [28, 29, 30, 31, 32, 33, 34]. By using this experience on different test rigs, the design of a new test rig tailored on strip dampers is here proposed.
The test rig allows measuring, at the same time, the frequency response of a blade with a strip and the contact hysteresis cycles desired for the direct determination of contact parameters.

briefly present the numericallycalculated nonlinear dynamic response of blades with strip dampers to highlight its strong dependence on the contact parameters and, consequently, the need for an accurate estimate of their values;

disclose the design of a novel test rig where both the frequency response function of the blade and the contact hysteresis cycles at the stripblade contact can be measured;
 prove the feasibility of the rig by testing its key components:

the systems to measure the contact forces and displacement at the contact;

the loading system to simulate the centrifugal force on the strip.


guaranteeing a uniform pressure distribution across the contact surface;

avoiding contact with the strip damper since, as will be shown below, traditional contact loading systems change the surface contact conditions.
Dynamic Equilibrium Equations

\(M, C,\) and K are the mass, damping and stiffness matrices of the system respectively. They are obtained by combining the mass, damping and stiffness matrices of both strip and blades. These matrices are extracted from the FE models of strip and blades after a Craig BamptonComponent Mode Synthesis (CBCMS) reduction [21].

\(\textbf {x}\) is the vector of the DOFs of strip and blades. Due to the CBCMS reduction, this vector includes both modal DOFs and the physical DOFs which are the displacements of the master nodes retained in the reduction.

\(\textbf {F}_{\textbf {ex}}\) is the vector of external excitation acting on the blade airfoil, see Fig. 1(b).

\(\textbf {F}_{\textbf {c}}\), is the vector of centrifugal forces pushing the strip against the blade platform, see Fig. 1(b).

\(\textbf {F}_{\textbf {nl}}\) is the vector of friction nonlinear forces generated between the coupled nodes stripblade by their relative displacements. Eq. 1 is nonlinear due to the presence of \(\textbf {F}_{\textbf {nl}}\) which depends on the relative displacement and velocity between strip and blades.
Equation (1) is written in the time domain and it can be converted in the frequency domain and solved by the wellknown MultiHarmonic Balance Method (MHBM) [19, 20, 38, 39]. In this way the system of nonlinear second order differentiaequations (1) is turned into a set of nonlinear algebraic complex equations, obtaining sensible reductions in the computational times. This is possible because the external excitation \(\textbf {F}_{\textbf {ex}}\) is periodic (sinusoidal). Consequently, also the displacements \(\textbf {x}\) and the contact forces \(\textbf {F}_{\textbf {nl}}\) are assumed to be periodic and are approximated by Fourier series. The presence of frictioninduced nonlinearities will produce displacements and contact forces which are still periodic but not perfectly sinusoidal (hence the need for the MHBM, rather than the simple HBM, to ensure an adequate harmonic support).
Contact Model
The contact stiffness values assigned to each contact node (i.e. local contact stiffness, \(k_{n}\), \(k_{tx}\) and \(k_{ty}\)) depend on the contact stiffness of the interface and on the number of contact nodes chosen to represent it. For instance, for \(k_{n}\): \(k_{n}=k_{ng}/N_{c}\), where \(k_{n}\) and \(k_{ng}\) are the local and global normal contact stiffness respectively, and \(N_{c}\) is the total number of the couples of nodes which approximate the contact between strip and blade platforms. The same relationship holds for \(k_{tx}\) and \(k_{ty}\). If the contact state is ”stick”, the relation between forces and displacements at the contact is linear and governed by the contact springs described above. If the contact state is ”slip”, normal and tangential forces are linked by a proportionality constant which can be positive or negative depending on the direction of slipping (i.e. Coulomb’s law of friction applies, \(T=\pm \mu N\)). If the contact state is ”liftoff”, contact forces are zero as the contact nodes are separated.
Numerical Calculation Procedure

implementation of the CBCMS reduction technique to reduce the size of the FE model without loss of accuracy or relevant information [21];

implementation of a stateoftheart contact model, described in the previous section, to take into account the presence of friction. The contact model is implemented in the time domain to ensure an adequate representation of all contact situations (including slip and liftoff) and the AFT (Alternative Frequency Time) method [43] is applied to switch between frequency and time domain. Contact parameters (i.e. contact stiffnesses and friction coefficient) are needed as input by the code [41, 42];

presence of the analytical Jacobian computation in the iterative solver to ensure efficiency [20, 22, 44];

presence of the inhouse Jacobian Alert Algorithm ^{2} to ensure an accurate solution obtained in a timely manner [22].
The main output of the numerical tool is the forced response of the structure under different excitation levels, a vital indication in the bladestrip design process.
The present study is focused on the second mode of the system, i.e. the blade neckbending mode shape. The choice is motivated by the fact that this mode is largely influenced by the presence of the strip. In fact, the strip acts as a constraint under the blade platform thus affecting considerably the motion of the neck. On the contrary the first mode shape is pure airfoilbending and therefore the strip does not alter significantly the system behavior. This difference between the two modes is easily quantified by comparing the relative motion of the platform kinematics with that of the airfoil. The first pure airfoilbending mode sees a relative motion between neighboring platforms which is only 0.003 \(\%\) of the reference airfoil motion, while the second neckbending mode sees a relative platform motion which is 0.13 \(\%\) of the reference airfoil motion.
Figure 5(b) explores the influence of the friction coefficient, \(\mu \), on the FRFs for a case where the contacts slip during the period of vibration. Once again, it is observed that the predicted value of the FRF amplitude is different for the different \(\mu \) values. For a given value of excitation force (F_{ex} = 20 N in Fig 5(b)), assuming \(\mu = 0.6\) instead of \(\mu = 0.1\) can change the calculated amplitude of 50%.
The Need for a Direct Experimental Investigation
In the previous section, it was shown that the calculation of blades with strips can be performed using the wellknown existing numerical techniques tested on blades with solid dampers and shrouds. It was also shown that contact parameters values can change significantly the numerical prediction of the FRFs (see Fig. 5). The most challenging and pressing target, according to these authors, is the determination of the contact parameters (k_{n}, k_{t} and \(\mu \)) values to be given as input to the numerical tool.
The problem of the determination of realistic values of the contact parameters is well known in literature [12, 36] and it is an issue that it is not fully solved yet. Furthermore, to the authors’ knowledge, there are no existing investigations on the estimation of the stripblade contact parameters, and no direct assessments of the adequacy of the contact model in this novel context.

Modeling. The strip damper has a very low mass: as a consequence, the contact pressure induced by the centrifugal force is quite low (of the order of 0.3 MPa). Results available in literature obtained for solid dampers at realistic centrifugal forces are obtained at much higher contact pressures (1030 MPa is a typical range). As there is an evidence that contact parameters of conforming contacts are heavily influenced by contact pressure [14], values obtained for solid dampers may not be used for the flexible damper case.

Testing. The strip damper is very flexible: the test rigs found in literature for the determination of contact parameters [12, 24, 42] apply contact pressure on one of the contacting bodies by means of a set of wires and pulleys connected to dead weights. The strip damper cannot be loaded by wires due to its high flexibility. The wires would deform the strip thus producing unrealistic contact conditions. The appropriate loading system for a strip damper should simulate the uniform pressure distribution provided by the centrifugal force, possibly without any contact with the strip.
Definition of the Test Rig Requirements

the test rig should be nonrotating in order to allow the accurate measurement of forces and displacement at the contact. This implies that the centrifugal force, generated only in rotating conditions, must be simulated in a different manner;

the loading system simulating the centrifugal force on the strip should not be in contact with the strip in order to avoid influence on the strip stiffness and overall system dynamics;

the hysteresis cycles at the bladestrip contact should be measured directly in order to assess the adequacy of the contact model and ultimately derive its calibration parameters;

contact pads on the blade platform should be replaceable in order to be manufactured with different materials;

the blade should be excited with a stepsine excitation force with constant amplitude. Since the system is nonlinear, a precise knowledge of the amplitude of the force exciting the system is required.
The two black boxes M and L in Fig. 6 are key components that must be purposely designed. Box M is the measurement system of the hysteresis cycles at the bladestrip contact. This means that box M should include a system to measure the contact forces and a system to measure the tangential relative displacement between strip and blade platform.
Box L is the load system simulating the centrifugal force on the strip, this system should apply a pressure on the strip without direct contact.
 a
In configuration a) one blade coupled with one strip is tested and the hysteresis cycle can be directly measured on the right side where the strip is in contact with the box M.
 b
In configuration b) two blades with one strip between them are tested, in this case the measurement of the hysteresis cycles on the strip is no more possible, but the FRFs of the two blades in contact with the strip can be measured.
The Test Rig Mechanical Structure
The clamping system is shown in Fig. 7(a). A hydraulic piston moves a clamping head that constrain the item to be tested. In the present study a dummy blade is constrained, as shown in Fig. 7(b).
The dummy blade (Fig. 7(b)) is in onepiece with a prismatic basis where the clamping force is applied. The blade platform is designed with a replaceable contact pad (in contact with the strip) to test different materials.
The Contact Forces Measuring System
The system to measure the contact forces has been previously designed, set up and tested on a test rig for the direct experimental investigation of solid UPDs and their contact with the corresponding platforms [33, 34].
To practically obtain this high longitudinaltotransverse stiffness ratio, each leg of the L shaped is composed of two thin parallel strips as shown in Fig. 9(b). The two limbs are perpendicular to each other and their axes intersect in a point as close as possible to the contact surface. In [34] it was demonstrated that the force on transducer, \(R_{n1}= 0.995F_{1}\) and \(R_{n2}= 0.995F_{2}\). This accuracy can decrease to 0.991 if the intersection of the axes of the two limbs is at \(\pm \)1.5 mm from the center of the contact area.
The system with two limbs is applied to the strip test rig in configuration a), as sketched in Fig. 6(a). The system is oriented so that the normal (N) and tangential (T) components of the contact forces are separately detected by the two force transducers \(t_{1}\) and \(t_{2}\).
The Test Rig Assembly
The Measurement of the Relative Displacement
The Strip Loading System

to be noncontact since the strip is very flexible. Any contact device can deform the strip and modify the bladestrip contact surface;

to be capable of producing a uniform stripblade pressure of the order of magnitude of the pressure induced by the centrifugal force in service. A typical realistic pressure value on the contact surface stripblade is 0.3 MPa;

to be capable of guaranteeing a constant pressure for the time needed to measure contact hysteresis cycles and blade forced responses (about ten minutes).
The contact pressure is here measured by a LLLW Prescale Fujifilm film positioned between the strip and the support (see Fig. 12(b)). The measurement result is shown in Fig. 13(a). The value of the pressure on the contact area is obtained using the calibration scale provided by Fujifim. The color distribution is fairly uniform with darker spots, mainly present in the lower left corner. This darker region may be caused by local elastic deformations of the strip subjected to stronger air jets (the lower portion of the strip is also closer to the pressure inlet). The average pressure value is 0.28 MPa with minor deviations across the contact patch (≈ 77 \(\%\) of the area has a local pressure value in the [0.250.31] MPa range). Since this contact pressure is slightly lower than the target (0.36 MPa) the authors deemed useless performing tests at lower inlet pressure values. Achieving higher contact pressure values is possible by installing an inlet pressure increaser. Thanks to the pressure regulator this pressure value at the inlet can be kept constant for more than ten minutes, the time required for the measurement of the contact hysteresis cycles and of the blades forces response.

it is in contact with the strip modifying the strip stiffness and the contact area;

it does not apply a uniform pressure since, as shown in Fig. 13(b), the pressure is applied mainly on the strip edges with pressure peaks \(>0.7\) MPa.

does not touch the strip, and therefore does not influence the strip stiffness and contact area;

is able to provide a uniform and continuous pressure at the bladestrip contact, with a realistic pressure value (about 0.3 MPa).
Determination of Contact Parameters and Test of the Force and Relative Displacement Measuring Systems

friction coefficients will be obtained through the analysis of the Tangential/Normal force ratio (see Fig. 14(a)). When the T/N signal is constant in time and equal to a maximum the contact is in slip and \(\mu =\pm T/N\) where \(\mu \) is the friction coefficient according to Coulomb’s definition. If T/N is instead varying in time then the contact is in stick.

Tangential contact stiffness values will be estimated using the hysteresis cycle (Fig. 14(b)) which relates the relative tangential displacement at the contact to the corresponding component of the contact force T. The slope of the hysteresis (obtained during a portion of the period when the T/N ratio is varying, i.e. stuck contact) represents the tangential contact stiffness \(k_{t}\).
Conclusions
The paper goal is to offer a solid contribution in the field of flexible damper testing. This subject is still largely unexplored, but nevertheless essential to trustworthy predictions of the nonlinear forced response of turbine blades.
It is here shown that the computed frequency response of two blades with a strip damper strongly depends on the chosen contact parameters values (friction coefficient, normal and tangential contact stiffnesses) at the stripblade contact. Testing flexible dampers in a controlled environment (e.g. nonrotating rig) to the purpose of estimating contact parameters is challenging, as it requires providing a uniform centrifugal load on the strip without modifying its stiffness and contact conditions.
The design of a novel test rig tailored on strip dampers is here proposed. The test rig design includes a measurement system for contact forces (1 \(\%\) accuracy) and for relative displacements at the contacts (0.08 \(\mu m\) accuracy) which was already tested on a previous rig for solid UPDs.

it produces an average pressure across the contact surface up to 0.28 MPa, which is a realistic value of the pressure on the strip in service during rotation;

the pressure distribution is uniform, in fact 77 \(\%\) of the contact area shares a local pressure in the \(\pm 10\%\) range about the average value;

it provides a pressure which is constant in time thanks to a pressure regulator of the compressed air at the inlet;

unlike classical punch and dead weight systems, it does not modify the contact conditions nor the strip stiffness.
Footnotes
 1.
The simulation of the full bladed disk is easily obtained by imposing the wellknown cyclic symmetric boundary conditions [35].
 2.
The Jacobian Alert Algorithm allows the user to run the MHBM simulation with a small harmonic support and gives a warning (hence the term ”Alert”) only if the error grows above a userdefined threshold, thus prompting the user to increase it to ensure accurate results. It is a cheap and effective alternative to standard convergence studies, and essential whenever severe nonlinearities are present (e.g. slipping or liftoff of contact points).
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