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Daniele Botto1
Department of Mechanical
and Aerospace Engineering,
Politecnico di Torino,
Corso Duca degli Abruzzi 24,
Torino 10129, Italy
e-mail: daniele.botto@polito.it
Chiara Gastadi
Department of Mechanical
and Aerospace Engineering,
Politecnico di Torino,
Corso Duca degli Abruzzi 24,
Torino 10129, Italy
e-mail: chiara.gastaldi@polito.it
Muzio M. Gola
Department of Mechanical
and Aerospace Engineering,
Politecnico di Torino,
Corso Duca degli Abruzzi 24,
Torino 10129, Italy
e-mail: muzio.gola@polito.it
Muhammad Umer
Department of Mechanical
and Aerospace Engineering,
Politecnico di Torino,
Corso Duca degli Abruzzi 24,
Torino 10129, Italy
e-mail: muhammad.umer@polito.it
1
An Experimental Investigation
of the Dynamics of a Blade With
Two Under-Platform Dampers
Several experimental apparatuses have been designed in the past to evaluate the effectiveness of under-platform dampers. Most of these experimental setups allow to measure
the overall damper efficiency in terms of reduction of vibration amplitude in turbine
blades. The experimental data collected with these test rigs do not increase the knowledge about the damper dynamics, and therefore, the uncertainty on the damper behavior
remains a big issue. In this paper, a different approach to evaluate the damper–blade
interaction has been put forward. A test rig has been purposely designed to accommodate
a single blade and two under-platform dampers. One side of each damper is in contact
with a ground support specifically designed to measure two independent forces on the
damper. In this way, both the normal and the tangential force components in the
damper–blade contact can be inferred. Damper kinematics is rebuilt by using the relative
displacement measured between damper and blade. This paper describes the concept
behind the new approach, shows the details of the new test rig, and discusses the blade
frequency response from a new point of view. [DOI: 10.1115/1.4037865]
Introduction
Devices capable of dissipating energy through friction are commonly used in turbines to reduce the vibration amplitude of the
blade in resonant conditions. Among the different devices used in
the aerospace industries, under-platform dampers are widely used
in turbo engines. Despite their simplicity, the simulation of the
under-platform damper behavior is not an easy task, and engineers
have been working in order to improve the accuracy with which
theoretical contact models predict the damper dynamics.
Several test rigs were developed in the past to experimentally
investigate the effect of under-platform dampers on blade dynamics. In Ref. [1], an experimental apparatus consisting of a single
blade/single damper was developed to measure the damper performance in terms of vibration stress reduction. The experimental
results were used to assess the capability of a new contact
model [2].
Nowadays, one of the most common test rig architecture is
composed by one damper placed between two blades excited with
a shaker [3–6]. In Ref. [7] and later in Ref. [8], the measurements
of the damper rotation were added to the standard response
function measurements. A modified architecture was used in
Refs. [9,10] in which two dampers were in contact with the different platforms of the same test blade. The other side of the damper
was in contact with a more rigid structure called dummy blade.
Moreover, the test blade was excited by a pulsating air jet. In all
these experimental setups, the centrifugal load acting on the
1
Corresponding author.
Contributed by the Structures and Dynamics Committee of ASME for publication
in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received
July 13, 2017; final manuscript received July 20, 2017; published online October 17,
2017. Editor: David Wisler.
damper was simulated by dead weights attached to the damper
through wires and pulleys arrangement or solid strips. In a more
complex test rig [11], a 24-blade assembly was excited with a
rotating force to investigate the damper behavior at different nodal
diameters of the disk. Dampers were loaded with dead weights as
well. Dampers are loaded in a more realistic way if tests are performed using rotating disks. In Ref. [12], an experimental and
numerical study has been performed on a thin-walled damper in a
rotating disk. Blades were excited with piezoelectric actuators. In
Ref. [13], the excitation force was simulated by employing a permanent magnet fixed to the rig.
All the previously cited experimental setups mainly study the
overall effect of the damper on the blade in terms of vibration
amplitude reduction and resonant frequency shift. This black-box
like approach is worthwhile to evaluate the capability of the
damper to reduce resonant displacements but it does not allow a
better understanding of the behavior of the damper. These test rigs
are not capable of analyzing the dynamics of the damper in depth,
nor its kinematics in terms of damper/under-platform relative
displacement.
On the other side, a number of test rigs [14–18] were developed
to measure the contact parameters, namely friction coefficient and
tangential contact stiffness, in controlled laboratory conditions.
Although the normal and tangential contact stiffness of a complex
geometry can be estimated by simulating the contact as in Refs.
[19,20], an experimental measurement of these parameters is the
most reliable option. The need for determination of the contact
parameters in working conditions close to reality led to the first
damper test rig developed at AERMEC laboratory.2 This rig was
built in 2008 [21] and was a first step toward a deep investigation
2
http://www.aermec-dimec.polito.it/
Journal of Engineering for Gas Turbines and Power
C 2018 by ASME
Copyright V
MARCH 2018, Vol. 140 / 032504-1
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Table 1
Rig
Scope
Input
Motion
Blade response
Frequency range
Contact pressure
Damper test rigs capabilities
2
Piezo damper
Resonant blade/damper
Stand-alone damper
Piezo
Any in-plane
no
Up to 160 Hz
Up to 2 MPa
Coupled blade/damper
Blade kinematics
Constrained by blade
yes
Blade resonance (300–2000 Hz)
Up to 6 MPa
of dampers behavior. Since then, the test rig has been used to
investigate the behavior of several dampers in terms of kinematics
and force transmission characteristics [22,23]. A numerical model
of the damper/test-rig system was first presented in Ref. [24],
together with the first version of the contact parameters estimation
procedure, subsequently improved in Ref. [25].
This first damper rig and the damper þ blade rig presented here
share the same ultimate goal: measuring the hysteresis cycle
produced by the damper between the platforms. However, their
structure and working principles are different. In the first rig [21],
the trajectory and amplitude of the input motion can be finely controlled by means of two piezo actuators connected to a dummy
platform (no blades are present). However, the frequency range
the rig can explore is limited (lower than 160 Hz). In the rig
presented in this article (see Sec. 2), the motion is achieved by
exciting a blade at resonance with an electromagnetic shaker;
therefore, the variety of platform kinematics the user can investigate is limited. However, the rig is excited in the frequency range
actually encountered by the blades mounted on a real working
engine. Furthermore, in this second rig, it is possible to directly
measure the effect of the damper hysteresis cycle on the blade
amplitude of vibration.
These two test rigs have complementary capabilities, as summarized in Table 1. Furthermore, they offer the possibility to test
the same damper and compare the resulting performance in terms
of hysteresis cycles and contact parameters. Section 4 contains
such a comparison for the pre-optimized damper geometry
presented in Ref. [25].
Fig. 1
032504-2 / Vol. 140, MARCH 2018
Test Rig Description
The test rig has been developed to fulfill the following
specifications:
(1) The architecture of the dynamic system must allow
measurements of forces on the contact surface and their relative displacements. To this purpose, the most suitable
architecture is then a single blade in contact with two
dampers. On one side, the dampers are in contact with the
blade under-platform while on the other side, they are in
contact with pads placed on a ground platform.
(2) The contact pads must be replaceable. Real dampers and
blades are manufactured in a variety of materials and
geometries and the test rig must be able to investigate
damper–blade pairs with different materials and contact
geometries. Replaceable contact pads provide cost advantage if compared to the replacement cost of the whole
ground platform.
(3) The rig must allow testing different blades with minor
adjustments to the apparatus, provided the maximum blade
size is not exceeded.
(4) A regulated and measurable clamping force is required to
apply a load on the blade which simulates the effect of the
real centrifugal load while the turbine runs.
(5) The designed test bench must be capable of measuring the
damper contact forces on the ground platforms.
(6) The test rig must allow measuring the relative displacement
between the contact surfaces.
Figure 1 shows the top view, a section, and two details of the
rig. The test rig is composed of three main subassemblies, namely
a central block and two lateral blocks. The central block is made
of two symmetric parts, (1A) and (1B). The two symmetric parts
are fixed to an optical table through a base plate (2) with 32 vertical bolts. The two parts house a clamping mechanism. The clamping mechanism applies a force simulating the centrifugal load on
the blade. This clamping mechanism is made of two wedge
blocks, the lower wedge (3) and the upper wedge (4), with a slope
of 1:10. These wedges convert a force FB applied perpendicular to
the longitudinal axis of the blade into a pushing force FP acting
along the longitudinal axis. The force FB is developed by
Overview of the test rig and details
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tightening the main bolt (5) on the bottom wedge (3). Due to the
1:10 slope between the two wedges, the nominal force amplification Anom between the bolt force FB and the pushing force FP is
ten in ideal, frictionless conditions. A strain gages-based load cell
(6) is placed between the main bolt (5) and the lower wedge block
(3). This load cell measures the bolt force FB. A thrust ball bearing (7) is placed between the main bolt and the load cell to allow
their relative rotation. The subassembly composed of the main
bolt, the thrust ball bearing, and the load cell is enclosed in a casing (8). The casing is tightened with six screws to the central
block. Two rails of linear flat roller bearings (9) are introduced
between the moving contact surfaces to minimize the friction
losses. The first roller bearing rail is inserted between the lower
wedge (3) and the bottom fixed block (10). The second roller bearing rail is located between lower and upper wedge. However, it is
not possible to eliminate all friction losses in the clamp mechanism, and rolling friction still causes a small loss in the clamping
force. The actual clamping force FP,act acting on the blade root
and the actual amplification factor Aact can be calculated with the
following formulae,
FP;act ¼ FB
Aact ¼
cos a 2l sin a l2 cos a
sin a þ 2l cos a l2 sin a
FP;act
cos a 2l sin a l2 cos a
¼
;
sin a þ 2l cos a l2 sin a
FB
(1)
(2)
where l and a are the rolling friction coefficient of the linear flat
roller bearings and the slope between two wedges, respectively.
The clamp efficiency gc ¼ Aact =Anom is defined as the ratio
between the actual and the nominal amplification factor. The friction coefficient l of the linear flat roller bearings ranges between
0.001 and 0.0015 [26] and gives an efficiency of the overall
clamping mechanism gc 97%. The pushing force FP is transmitted to the blade (11) through the pushing block (12). The calculations of the state of stress on the blade root at maximum clamping
force were done during the design process. The ideal stress does
not exceed the strength of the blade material. An aligning pin (13)
allows small rotation between the pushing block and the upper
wedge, so that small misalignments can be recovered and a more
uniform pressure can be applied to the blade root. To avoid damage due to the high contact pressure, the self-aligning pin and its
counterparts on the pushing block and the upper wedge have been
designed with conforming contact surfaces. Pressure distribution
on conforming contact surfaces has been deduced by the graphs
found in Ref. [27]. The blade is inserted in the blade adapter (14)
in which a bucket groove has been machined to match the blade
root geometry. The blade adapter is pushed against the shoulders
on the central block. As the blade adapter is a replaceable part,
blades with a different root can be tested by simply changing the
adapter according to any blade root geometry. Eight long stud
bolts are inserted across the two parts of the central block, by
means of through-holes (15), to hold the complete assembly altogether and stiffen the structure. These stud bolts co-operate with
the vertical bolts to increase the rigidity of the overall structure.
Measuring the contact forces acting on the dampers is a significant feature of this novel test rig. The contact force measuring
device is visible in Fig. 2. The structure of this system is composed of two arms rigidly connected at one end. The axes of the
two arms are oriented at 90 deg to each other, and they intersect
each other at the point of the nominal contact between the damper
and the pad. Forces N and T are the normal and tangential contact
force, respectively. Two piezoelectric load cells (LC) are
coaxially mounted at the other end of both arms. The ratio
between the longitudinal and transverse stiffness of each arm is
about 100, so that the transverse component experienced by the
load cell can be neglected. The contact forces N and T are deduced
by the signals from the two load cells, namely R11 and R13 in
Fig. 2, and the geometry of the contact. The selected load cells
Journal of Engineering for Gas Turbines and Power
Fig. 2 View of the subassembly allowing the measurement of
the contact forces. R11 and R13 are the forces measured by the
load cells LC11 and LC13, respectively.
work with a charge amplifier with a low time-drift factor (i.e.,
lower than 5 mN/s). This feature allows to measure both static and
dynamic components of the contact forces.
The dampers, placed between the blade under-platforms and
the pads of the ground platforms, are loaded by a set of dead
weights simulating the centrifugal force (CF). The wire holes are
properly oriented and pass through the dampers center of mass:
their sections shrink down to match the wire diameter in correspondence of the center of mass to exactly reproduce the loading
conditions encountered on a bladed disk.
3 Testing Conditions, Observed and Derived
Quantities
Experimental tests were performed on a dummy blade with a
fir-tree root. In all the tests, the blade was clamped with a constant
radial force of 50 kN, a realistic value for similar blades in real
turbine. Tests were carried out with different centrifugal force on
the dampers and different exciting forces on the blade. For each
testing condition, a standard force-controlled frequency response
function (FRF) was performed. The acceleration at the tip of the
blade (normalized by the excitation force amplitude jFE j) is plotted as a function of the excitation frequency. An example of these
diagrams can be found in Figs. 3(a) and 3(b). It is well known that
the shape, position, and amplitude of the FRFs peaks unveil the
presence of a nonlinearity induced by friction; however, FRFs
alone do not give an insight into the damper behavior. The present
test rig is capable of linking a given point on a FRF to the
corresponding damper-platform contact forces and relative displacements. In the following subsections, all measured and reconstructed quantities are listed and described. This set of quantities
will then be used in Sec. 4 to analyze the dampers’ and the blade’s
behavior together with their mutual influence.
3.1 Observed Force Components. The readings of the load
cells mentioned in Sec. 2 and shown in Figs. 2 and 4(a) give the
complete in-plane force components at the cylindrical side of the
dampers. As already mentioned, the measurement chain composed of load cell and a proper charge amplifier allows measuring
the static component of forces as well as their dynamic variation.
The static components are necessary in order to reconstruct the
damper equilibrium (see Sec. 3.2) and to estimate the friction
coefficients.
The error introduced by the load cells time drift factor, albeit
low, must be controlled and minimized. The authors have therefore devised and followed the measurement protocol below:
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Fig. 4 The test rig setup (a) with measured and applied forces,
the laser setup (b) to record the blade platform in-plane motion,
and the laser setup (c) to measure tangential relative displacement at the flat-on-flat and cylinder-on-flat contacts
Fig. 5 Static equilibrium of the forces on the damper
(4) At the end of the session, the dampers are removed.
(5) The load cells, although completely unloaded, will display
a nonzero reading, due to the time drift.
(6) These nonzero readings are used to perform a correction on
the static values of the recoded forces. The drift is assumed
to be growing linearly with time.
Calculations shown that drift causes a loss of signal corresponding to 2 mN/s. Therefore, in a typical measurement lasting less
than 5 min the correction is negligible.
Fig. 3 FRF for (a) the free blade without dampers and (b) with
dampers loaded with CF 5 4.6 kg, (b) different forcing levels jFE j
have been investigated, and (c) measured and simulated hysteresis loops at the cylindrical side of the damper, corresponding
to the working conditions highlighted in Fig. 3(b)
(1) Before loading the dampers the load cells reference is set to
zero.
(2) The dampers are positioned and loaded.
(3) Measurements are carried out and forces are recorded (each
individual measurement is assigned the time at which it
was performed).
032504-4 / Vol. 140, MARCH 2018
3.2 Derived Force Components. The measured forces (R12
and R14, R11, and R13) are then rotated to compute the components
tangential and normal to the contact, here termed Tcyl and Ncyl
(see Fig. 5).
Furthermore, the damper static equilibrium is reconstructed by
neglecting damper mass inertia forces (lower than 0.1 N in the frequency range of interest) and therefore assuming contact and centrifugal forces to pass through one point, as described in Ref. [24]
and shown in Fig. 5. In this way Nflat and Tflat and their point of
application on the damper are determined. A sample result of this
computation can be found in Fig. 6(c).
3.3 Observed Kinematic Quantities. As shown in Fig. 4(c),
the relative displacements at the damper-platforms contacts are
measured by means of the differential laser head. These signals
will be plotted against the tangential component of the corresponding contact force in order to obtain the hysteresis at the
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3.4 Derived Kinematic Quantities. The blade platform can
be safely considered as a rigid body, given its bulky structure.
This allows deriving the platform movement using the instantaneous center of rotation (ICR) concept. The planar displacement of a
body is defined by the combination of a planar rotation and planar
translation. For any planar displacement, there is a point in the
moving body that is in the same place before and after the displacement. This point is the ICR, and the displacement can be
viewed as a rotation around this pole.
The present blade (see Fig. 4) was designed to be perfectly
symmetric; furthermore, its loading configuration is symmetric as
well. It is therefore reasonable to assume that the ICR will lie on
the blade axis. Using the platform rotation signal b and the horizontal displacement of point P uP, it is possible to derive the planar displacement of point P, here termed dP (see Fig. 7). Starting
from this vector, it is easy to determine the vertical position of the
ICR along the blade axis. Once the position of the ICR is known,
the displacement of all points of interest belonging to the blade
platform (i.e., contact point C in Fig.7(c)) can be derived. A few
examples can be found in Fig. 7. Furthermore, it will be shown in
Sec. 4.2 how the presence of the damper affects the position of the
ICR.
4
Results and Discussion
The response recorded at the tip of the blade is highly influenced by the presence of the damper and of the friction-induced
nonlinearities. As shown in Fig. 3(a), the damper shifts the first
bending mode resonant frequency from 410 Hz to 567.5 Hz in full
stick condition (jFE j ¼ 1 N) and lowers the force-normalized
blade amplitude of motion of one order of magnitude. Generally
speaking, as the forcing function increases, the amplitude of the
force-normalized peak tends to decrease, and its frequency to shift
to lower values (e.g., the peak at jFE j ¼ 50 N is down to 530 Hz).
This effect is produced by the dampers, which start to slip against
the platforms and
dissipate energy, therefore reducing the amplitude of the
FRF peak and
reduce the stiffness introduced by the contact in full stick.
Fig. 6 (a) Comparison of FRF and (b) hysteresis loops at the
damper cylindrical contact with different initial conditions, and
(c) Damper force equilibrium; force scale shown in the figure. In
the comparison, the dampers were loaded with CF 5 4.6 kg, and
the blade was excited with a force jFE j 5 20 N.
contact. Furthermore, the in-plane kinematics of the blade underplatform is reconstructed from measured data:
The platform rotation b ¼ DwP =Dx is measured by means of
a laser differential measurement (Fig. 4(b)).
The platform horizontal displacement uP (Fig. 4(b)) is
obtained using a single laser head.
Journal of Engineering for Gas Turbines and Power
4.1 Measured Contact Hystereses Explain Puzzling
Measured FRFs. A general understanding of the damper effect
on the blades is nowadays common knowledge in the turbine
dynamics field. However, there are still a series of open questions
which turbine designers and experimenters face.
The first question is related to the experimental characterization of damper–blades systems. Frequency Response Functions
(FRFs) sometimes produce unexpected results which are hard to
understand and justify. For instance, with reference to Figs. 3,
8(a), and 8(b), it can be noticed how for jFE j ranging from 1 N
to 50 N, as expected, the amplitude and the frequency of the
peak decrease. However, for jFE j 80 N the peak amplitude
and frequency start increasing again. This behavior is repeatable
and is observed at different loads on the damper. Without an
insight on the damper behavior, this phenomenon would remain
unexplained.
Thanks to the test rig capabilities, it is now possible to relate
the FRF shape to the corresponding damper behavior. Figure 3(b)
shows the hysteresis cycle at the cylindrical contact for two different values of jFE j (50 N and 100 N, respectively). It can be seen
that at jFE j ¼ 50 N the cylindrical interface is in microslip/at the
onset of gross slip: this behavior is easily simulated with the
numerical code described in Sec. 3.4. At jFE j ¼ 100 N, the contact
is slipping and therefore dissipating energy; however, during part
of the cycle, the damper appears to be “glued” to the platform.
Specifically, the tangential force changes its value, but no relative
movement between damper and platform is recorded. This behavior introduces an additional stiffness to the system, thus producing
an increase of the peak frequency.
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Fig. 7 Reconstruction of the in plane motion of the blade: (a) free blade with excitation force jFE j 5 5 N, (b) damper loaded with
CF 5 4.6 kg and jFE j 5 5 N, and (c) damper loaded with CF 5 4.6 kg and jFE j 5 100 N (c)
Fig. 8 FRF of the blade-dampers system: (a) dampers loaded with CF 5 2.6 kg and (b) dampers loaded with CF 5 6.6 kg (b)
Another experimental observation which puzzles designers during testing is the notable lack of repeatability for the same test
conditions. The results reported in Figs. 3 and 8 are obtained starting from jFE j ¼ 1 N and going up to jFE j ¼ 100 N. If the same set
of experiments is repeated, without unloading the dampers, in the
opposite order (from jFE j ¼ 100 N to 1 N) the FRFs obtained in
the microslip region (jFE j 20 N) will be markedly different
from the first set. An example can be found in Fig. 6. The result of
several test campaigns is reported in Fig. 6(a) to show that the
results are not randomly distributed. They are in fact quite repeatable, provided that the initial conditions are the same.
In the microslip regime, the steady-state solution is nonunique,
and it depends on the initial conditions, namely the value of the
static component of contact forces. Yang and Menq [28] pointed
out that, for the same input motion, “different initial states and
initial values of the friction forces at the beginning of simulation
may result in different friction force trajectories when their steady
states are reached.” More recently, the authors [29] showed that
using the damper static balance equations to compute the normal
preloads acting on the damper sides does not allow calculating a
unique solution if the contact is partially stuck. The authors
numerically demonstrated the effect of this under-determinacy on
under-platform dampers in Ref. [23]. Using the test rig presented
here, the numerical observations described earlier have found an
experimental counterpart. Figure 6(c) shows that the static component of the normal contact forces undergo a 50% increase if the
damper enters the gross-slip regime (i.e., if the blade is excited
with jFE j ¼ 100 N). This produces different steady-state hysteresis cycles (Fig. 6(b)): specifically if the static normal components
of contact forces are low, the damper is in microslip (purple solid
line). If the normal components increase, the steady-state cycle
will be larger (gray dashed line) but the damper will also remain
glued to the platform for a portion of the cycle, similar to what
was happening at jFE j ¼ 100 N (see Fig. 3(b)). The damper
032504-6 / Vol. 140, MARCH 2018
behavior at the contact is perfectly compatible with the FRFs: the
one corresponding to higher contact forces displays a higher peak
frequency (due to the additional stiffness) and a larger amplitude.
Apart from the theoretical explanation behind the discrepancy
between FRFs obtained under the same nominal test conditions,
the authors believe that identifying its cause is a valuable piece of
information to consider when performing experimental identification of friction-damped structures.
4.2 Comparison of Damper Parameters and Performance
on Two Independent Test Rigs. Another critical point when
designing damper–blade systems is the capability of performing
trustworthy and predictive numerical simulations. Recent results
on under-platform dampers have shown that the numerical capabilities now available (e.g., multi Harmonic Balance Method,
Analytical Jacobian Computation, etc) are up to the mark, provided that friction parameters necessary to contact models are
properly estimated.
To this purpose, the authors developed a contact parameter estimation technique [24], further improved in Ref. [30], based on the
experimental observation of the piezo damper only test rig.
Specifically:
friction coefficients are obtained by plotting the tangential/
normal force ratio at the contact. When the ratio is constant
in time and equal to a maximum, the corresponding friction
coefficient is set equal to the force ratio.
tangential contact stiffness values are derived from the slopes
of the portions of the hysteresis cycles at the contact corresponding to a stick state (varying tangential/normal force
ratio).
This technique has been applied to the data obtained presented
in this paper from the new blade þ damper resonant test rig. The
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Fig. 9(b)) have been marked by a symbol and a number repeated
on the corresponding points on the T/N diagram (e.g., Fig. 9(a)):
they are useful to guide the analysis of the cycle by cross comparison. From the combined analysis of the T/N diagram and of the
hysteresis at the cylindrical contact in Fig. 9, it holds
from markers 1 to 2, the cylindrical contact is sliding since
the Tcyl =Ncyl is constant and equal to a maximum, in that
case lcyl ¼ 0:6, exactly the same value measured on the
damper-only test rig [25];
from markers 2–3 and 5–1, a sharp increase of tangential
force without any associated movement is recorded;
from markers 2–5, the state is stick, the measured tangential
stiffness ktcyl ¼ 35 N/lm 6 9 N/lm, perfectly compatible
with the 30 N/lm 6 7 N/lm measured on the damper only
test rig [30].
The friction coefficient at the flat-on-flat contact lflat cannot be
estimated because the flat damper interface never reaches the
gross-slip condition (as shown by the sinusoidal shape of the
Tflat =Nflat ratio). Furthermore, the value of tangential stiffness at
the contact ktflat, in the 20–70 N/lm range, although compatible
with those recorded on the damper test rig [30], is here not shown.
The amplitude of Dtflat is lower than 0.1 lm in all investigated
cases, a value which is too low to allow a reliable determination
of ktflat.
The damper analyzed in this paper is the result of a preoptimization process, carried out in Ref. [25], to select a damper
geometry capable of reaching a generalized slip situation (all
interfaces are in gross slip). The damper was tested on the
damper-only test rig under a pure in-phase motion (e.g., pure vertical relative motion of the platforms) and verified the design
condition.
The generalized slip situation, however, is not verified on the
blade þ dampers test rig as demonstrated in Fig. 9(a). Since the
recorded contact parameters match those obtained on the damper
only test rig, the reason behind the different performances must be
caused by a different platform input motion.
4.3 Blade Platform Motion Reconstruction. The following
holds for the single test-rig blade-platform between groundplatforms where the absolute motion represents the relative
motion of two real-case adjacent blades. The in-plane blade platform input motion can be reconstructed using the procedure highlighted in Sec. 3.4. The results are reported in Fig. 7.
Analyzing the results of Fig. 7, it is possible to draw the following caveats:
Fig. 9 (a) Tangential over normal force ratio, (b) hysteresis
loop at the cylinder-on-flat contact (excitation force jFE j
5 100 N; dampers load CF 5 4.6 kg), and (c) Comparison
between the slopes of the hysteresis loops at the cylinder-onflat contact obtained on two different and independent test rigs
results of the contact parameter estimation procedure have been
compared to those obtained in Refs. [25] and [30] since, in these
works, the exact same damper has been tested.
A case with jFE j ¼ 100 N (similar to the one in Fig. 3(b)) has
been selected in order to ensure the cylindrical contact to
reach gross slip. Specific points on the hysteresis diagram (e.g.,
Journal of Engineering for Gas Turbines and Power
The position of the instantaneous center of rotation identifies
the mode of vibration of the blade, e.g., for a purely in-phase
motion (vertical displacement of the platform) the ICR lies
inside the platform itself, for a purely out-of-phase motion
the ICR vertical position moves infinitely downward the center of the hypothetical bladed disk.
The free cantilever blade investigated in this paper displays
an ICR in an intermediate position, specifically it lies in the
middle of the “neck” of the blade considering the blade root
not shown in Fig. 7.
The presence of the damper (see Fig. 7(b)) has a double
effect on the platform in-plane motion: on the one hand, the
amplitude of motion dramatically decreases, and on the
other, it shifts the ICR upward, thus modifying the direction
of motion as well as the amplitude.
The forcing level jFE j affects the amplitude of motion but
does not modify the position of the ICR (and therefore the
motion direction), as observed by comparing Figs. 7(b) and
7(c).
The procedure described in Sec. 3.4 has been applied to the
case reported in Fig. 7(c) to reconstruct the in-plane motion of
points belonging to the contact patch (point C). The resulting
MARCH 2018, Vol. 140 / 032504-7
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can, however, strongly influence the damper performance. Specifically in the case examined here, the platform input motion, reconstructed through a purposely developed technique, is almost
orthogonal to the flat-on-flat interface, which, as a result, fails to
reach the gross slip condition. This hypothesis is confirmed by the
results of a numerical model representing the damper between a
set of platforms.
These results highlight the importance of considering the blade
mode shape to achieve a full understanding of the system dynamics and as a mean to achieve a better and faster damper shape
optimization.
Fig. 10 Numerical model of the damper
Fig. 11(a) Measured and (b) simulated contact forces on the
damper for CF 5 4.6 kg and jFE j 5 100 N
displacement dC is almost orthogonal to the flat-on-flat contact; it
is therefore not surprising that the flat-on-flat contact fails to reach
the slip condition even at high forcing levels.
4.4 Damper Performance Simulation. The numerical model
used to simulate the damper behavior, fully described and validated in Ref. [24], is shown in Fig. 10. It represents the damper,
modeled as a rigid body, between two platforms, with assigned
motion. The contact is modeled through standard macroslip
elements.
Inputs to the numerical model are: the platform motion signal
dC reconstructed as in Sec. 3.4 and the friction contact parameters
estimated in Ref. [30] and confirmed by the measurements in
Sec. 4.2. The resulting force equilibrium is reported in Fig. 11(b).
It compares well with the measured counterpart (Fig. 11(a)), thus
confirming the soundness of the measurements and the interpretation of experimental results.
5
Conclusions
The capabilities of the novel test rig allow an insightful investigation into the damper behavior. Namely for each point on a FRF,
the corresponding hysteresis at the contacts, force equilibrium,
and platform kinematics can be produced. This set of diagrams is
particularly helpful in explaining unexpected phenomena (e.g.,
FRFs lack of repeatability), which often make experimental characterization of damper–blade systems difficult.
The test rig can be used to estimate friction contact parameters,
which compare extremely well with the values found for the same
damper on an independent test rig (damper-only test rig). These
findings speak for the soundness of both test rigs and for the estimation procedure itself.
The platform input motion produced by the blade mode shape
is quite different from the pure in-phase motion which was used
as a reference case on the damper-only test rig. It can therefore be
concluded that the platform motion (linked to the blades mode
shape) does not affect the values of friction contact parameters. It
032504-8 / Vol. 140, MARCH 2018
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