# Optimum design of damped vibration absorber for viscoelastic bladed disk assemblies.

1. Introduction

2. Formulation

Because of omitting the mistuning effects, an idealized perfectly tuned model of system, which contains of a cyclic chain of N identical, identically coupled blades that are uniformly attached to a rigid disk, is considered. Each blade is modelled as Euler-Bernoulli cantilever beam with constant material property and thickness. To represent the linear viscoelastic behaviour of the beam, Kelvin-Voigt model is chosen.

Transverse vibration of the beams is studied. According to Eq. (1) which introduces the relationship between the bending moment (M (x,t)) and deflected shape (w(x,t)) of the beam, equivalent flexural stiffness and damping constant of the beam is extracted:

P (M (x,t)) = - I ([?.sup.2]w(x, t)/?[x.sup.2]). (1)

P and Q are introduced as below:

P = [m.summation over (k=0)] [P.sub.k] [d.sup.k]/[dt.sup.k] and Q = [m.summation over (k=0)] [Q.sub.k] [d.sup.k]/[dt.sup.k]. (2)

Because of the harmonic characterization of applied load and according to Equation. (3) which is the complex compliance for Kelvin-Voigt model for the linear viscoelastic behaviour of materials:

J* (?) = 1/E + [µ.sub.?] i?. (3)

And due to the boundary conditions of the beam, after some analytical simplification, equivalent flexural stiffness and damping constant of viscoelastic beam is extracted as below:

[K.sub.e] = 3E[I.sub.b]/[L.sup.3] and [C.sub.e] = 3[µ.sub.?][I.sub.b]/[L.sup.3]. (4)

In the Eq. (4), [K.sub.e] and [C.sub.e] are the equal viscoelastic stiffness and damping parameters, E is the elastic modulus and [µ.sub.?] is the viscosity of the material, ? is the angular frequency, L is the length of the beam (effective length of the blade) and [I.sub.b] is its moment of inertia. In this paper, the model with one degree of freedom is considered for each blade. According to Fig. 1 the provided model is included N pendulums with the lengths of [L.sub.a] and the lumped masses of [M.sub.a] that are uniformly attached to the periphery of a rigid disk with a radius of H. The disk rotates with the constant speed of O around the axis through A .

With using Eq. (4), the flexural viscoelastic stiffness and damping of each blades are modeled with a linear torsional springs and dampers in the attachment point of the blade to the disk. Inter-blade coupling (due to the shrouds, aerodynamic damping and etc.) is modeled in the distances of ra and with linear springs and dampers .

The blade angle is shown with[?.sub.a]. For purely radial configuration of the blades, it is assumed that the springs and dampers are unstressed. Effect of rotational inertia is ignored and identical absorber that is modeled by a simple pendulum with the length of db and mass of ma are fitted in the each of the blades.

Damping of absorber is modelled using a torsional damper acts at attachment point of absorber pendulum to the blade (point P). [ß.sub.b] describes the angle of absorber which is relative to its corresponding blade.

The terms related to blade and absorber is introduced in Table 1.

Fig. 2 shows the blade and the absorber fitted in it, which is constituted the sector .

The kinetic energy of the entire system is extracted according to Eq. (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

Gravitational effects are ignored and due to the elastic inter-blade coupling and flexural stiffness of the blades, the system potential energy is given by Eq. (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

According to Eq. (7), the Rayleigh dissipation function describes the damping of the absorber and the inter-blade damping:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The blades are forced harmonically and in the transverse sense by engine order excitation of order n. The range of 0 < n < N is considered for the excitation order. The values of n = N, will not affect the results . The engine order excitation is modeled with the Eq. (8):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

In the above equation, [F.sub.o] is strength of excitation and [f.sub.i]. = 2p n/N (i-1) is inter-blade phase angle . Lagrange's method is employed and the equations of motion are extracted. To extract the dimension-less form of the equations, the time is rescaled according to t = [?.sub.o]t and [?.sub.o] = [square root of (K/[ML.sup.2]] is the undamped natural frequency of single isolated blade. Next, the dimension-less form of the equations are extracted according to the parameters that are defined in Table 2 and are linearized for small blades and absorbers motions.

In matrix-vector form, equations of motion for the 2DOF sector (blade with the absorber) are given by Eq. (9):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

Where the vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] captures the sector dynamic vector and the elements of the sector mass, damping and stiffness matrices are defined below:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

The inter-blade coupling stiffness and damping are captured as Eq. (13):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

The sector force vector is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

Each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is stacked into the configuration vector q = [([z.sub.1],[z.sub.2], ...,[z.sub.N]).sup.T] so the governing matrix equation of motions for overall system is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

Where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the overall system mass, damping, stiffness and force matrix respectively and in terms of Circulant operators, they are shown as below:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

By exploiting the system cyclic symmetry and the Circulant structure of its matrices, the steady state forced response of the overall system can be obtained with the use of modal transformation including the complex Fourier matrix. [11, 15] Therefore Equation. (15) which is the governing matrix equation of motion for the overall N-DOF system (N=2 for this investigation) is decoupled to N (N=2 for this investigation) reduced-order equations. The steady-state response of a system with 2N degree of freedom is converted to the solution of a single system with 2 degree of freedom under harmonic excitation. With the exception of the constant phase, which is transferred from each sector to another sector, the blades behavior is fully identical. The constant angle [f.sub.i] shows this difference. Note that it is shown if n (engine excitation order) is an integer, the system will only be excited at mode n + 1 .

3. Numerical results and discussions

In the following, elastic and linear viscoelastic behavior of the blade have been considered and compared via using parameters values, which are mentioned in Table 2. Equivalent viscoelastic parameters of the blade have been considered according to Eq. (4). First, the area under the frequency response curve of the single isolated blade without the absorber is represented in Table 3. (For this aim, parameters of the absorber are considered equal to zero)

It is clear that the area under system frequency response curve for the viscoelastic blade, which is demonstrated by [S.sub.VW], is smaller than [S.sub.EW] that is represented that area for elastic blade. For the viscoelastic blades, the dissipated energy reduced by 87% in relative to the elastic blades.

Design parameters include the mass, length of the pendulum of the absorber and its damping is determined optimally from the wide range of each parameters and with simultaneous use of [H.sub.2] optimization criterion  and common numerical methods. The aim of the [H.sub.2] optimization criterion is to reduce the total vibration energy of the system over all frequency. Reduction of the dissipated energy will prevent the failures like High Cycle Fatigue. Then with using of content of Table 3, the absorber is considered and amplitude of the frequency response curve of the system is illustrated in Fig. 3 as a function of rotor speed in dimensionless form. Table 4 shows the areas under the frequency response curve for Fig. 3. The viscoelastic blade can reduce dissipated energy by 89% in relative to the elastic one.

Fig. 4 shows the amplitude frequency response curve of the system with optimum values of absorber parameters for the viscoelastic and elastic blade respectively.

Tables 5 and 6 shows the area under the frequency response curve and optimum values of absorber parameters for the viscoelastic and elastic blade respectively.

According to Table 7, it is clear that the viscoelastic blades, even when they are not equipped with the vibration absorber, reduce dissipated energy more than the elastic blade (almost 87%) and even elastic blade which is equipped with optimum vibration absorber (almost 75%).

When the viscoelastic blades are equipped with optimum centrifugally driven order tuned vibration absorber, it is observed that the dissipated energy reduced 60%, 94% and 90% in relative to the viscoelastic blade without absorber, the elastic blade without absorber and the elastic blade with the optimum vibration absorber respectively. It is clear that the minimum area is related to the situation where the viscoelastic blade is equipped with optimum vibration absorber (row 4). Moreover, with the comparison of row 3 and 6 of Table 7, it is concluded that, optimum vibration absorber can reduce dissipated energy in the viscoelastic blade more than the elastic blade. (Up to 12%) Also with comparison of amplitude frequency response of Figs. 3 and 4, it can be concluded that the optimum vibration absorber can cause amplitude frequency response curve of viscoelastic blade to act smoother, especially in resonant frequency, than the elastic blade. In order to investigate the effect of values of absorber parameters on the system frequency response, varied amounts should be considered. For example, Table 8 shows the effects of the variation of absorber damping of the viscoelastic blade when other parameters are constant.

Table 9 shows that situation for the elastic blades. It is shown that the area under the curve of the system frequency response is changed. With the increasing the damping of the absorber, that is, by moving towards a direction in which, the absorber be locked in relative to the blade, the area under the system frequency response curve is remained constant. With the change of each other parameters, the area under system frequency response curve is deviated from its optimum conditions. It is found that the smallest area is related to the optimum values of the absorber parameters of the viscoelastic blade.

It is important to declare that to study the characterization of the elastic blade, the effects of viscosity of the material have been ignored. Hence for elastic situation, the blade lumped damping ([C.sub.a]) and its torsional damping in dimensionless form ([?.sub.a]) is equal to zero in all equations.

To emphasize the results of this investigation, the study of Olson  is examined. He considered the undamped absorber with fixed mass and with the parameter of ß as the absorber detuning parameter and extracted the system frequency response for different conditions. He also disregarded the effects of aerodynamic damping. Note that he had considered the elastic characterizations for both blade and disk. Definitions and values in  are assembled on the formulas and methods of this investigation and results are obtained and compared with results of in Fig. 5. Results are matched and accuracy of this investigation is approved.

5. Conclusions

Note that the designed absorber is only capable of removing one of the system resonances. Therefore, it seems necessary to do future works in which each blade is equipped with two or more absorbers. In this research, the absorber was moved in a circular path, so the investigation of the effects of the movement of the absorber under conditions in which it is moving in a desired way, is recommended. Although in relevant literature, some studies have been performed in which, the effects of impact absorber are investigated  but it seems necessary to conduct experimental studies to more accurately measure the validity of the results and to more accurately design absorber. Finally, the investigation of the effects of mistuning as an important factor in determining the behavior of the system and absorber is very important.

Accepted November 12, 2015

http://dx.doi.org/10.5755/j01.mech.21.6.13243

References

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[13.] Kaneko, Y.; Ohta, M.: Mori, K.; Ohyama, M. 2012. Study on Vibration Response Reduction of Bladed Disk by Use of Asymmetric Vane Spacing, International Journal of Gas Turbine, Propulsion and Power Systems, 4: 35-42. http://dx.doi.org/10.1299/kikaic.78.1398.

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*Department of Mechanical and Aerospace Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran, E-mail: a.a.ghaderi@gmail.com

** Department of Mechanical and Aerospace Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran, E-mail: a-mohamadzadeh@srbiau.ac.ir

*** Department of Mechanical and Aerospace Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran, E-mail: m-nikkhah@srbiau.ac.ir
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Table 1
List of variables and parameters of blade and absorber

Parameter Description Unit

H Radius of the disk m
[L.sub.a] Length of the blade m
[d.sub.b] Length of the absorber m
Distance from base point of blade to
[d.sub.b] inter-coupling elements attachment m
point
[m.sub.a] Absorber mass Kg
[k.sub.a] Stiffness coupling between sectors N/m
[c.sub.a] Damping coupling between sectors N.s/m
[C.sub.[bar.b]] Absorber damping N.s/m

Table 2
Descriptions and values of dimensionless
variables and parameters

Parameter Description Value

s = Angular speed [0-1]
O/[?.sub.o]

[µ.sub.a] = Lumped mass of the blade 0.245
[M.sub.a]/M

[µ.sub.[bar.a]] = mass of the absorber 0.0015
[m.sub.a]/M

[?.sub.a] = [L.sub.a]/M Blade pendulum length 1

[?.sub.[bar.a]] = Absorber pendulum length 0.03
[L.sub.a]/M

d = H/L Radius of the disk 0.72

[f.sub.a] = [F.sub.o]L/K Strength of e.o excitation 0.04

[?.sub.a] = 1/[L.sup.2] Blade torsional damping 0.3
[C.sub.a]/[square root of
M/K[L.sup.2]]

[?.sub.[bar.a]] = Aerodynamic damping constant 0.0002
[([r.sub.a]/L).sup.2]
[C.sub.a]/[square root of
M/K[L.sup.2]]

[?.sub.[??]] = Absorber damping constant [0-1000]
1/[L.sup.2] [C.sub.[bar.a]]/
[square root of
M/K[L.sup.2]]

[v.sub.a] = Blade torsional stiffness 0.0005
[square root of [K.sub.a]/K]

[v.sub.b] = [square root of Strength of the coupling 0.0003

[a.sub.b] Distance from middle of 0.87
absorber base point P

n Engine order excitation 6

Table 3
The areas under the frequency response curve

without vibration absorber

without vibration absorber)

Table 4
The areas under the system frequency response
curve related to Fig. 3

[S.sub.VV] (viscoelastic 0.0014

with vibration absorber)

Table 5
The areas under the system frequency response
curve for optimum values of absorber parameters

with optimum vibration absorber)

optimum vibration absorber)

Table 6

Optimum values of the absorber parameters
for the viscoelastic and the elastic blades

Value Viscoelastic Elastic

[µ.sub.[??]] 0.00108 0.00468

[?.sub.[bar.a]] 0.0251 0.0401

[?.sub.[??]] 0.00804 0.00461

Table 7
Ratio of the areas that are mentioned
in Tables 3 and 5

1 [S.sub.VW]/[S.sub.EW] 13%
2 [S.sub.VW]/[S.sub.EO] 25%
3 [S.sub.VO]/[S.sub.VW] 40%
4 [S.sub.VO]/[S.sub.EW] 6%
5 [S.sub.VO]/[S.sub.EO] 10%
6 [S.sub.EO]/[S.sub.EW] 52%

Table 8
Different values for the absorber damping while
other parameters are constant (viscoelastic blade)

[µ.sub.[??]] 0.00108 0.00108 0.00108
[?.sub.[bar.a]] 0.0251 0.0251 0.0251
[?.sub.[??]] 8*e-4 8*e-3 8*e-2
[S.sub.P] 0.000978 0.000807 0.000912

[µ.sub.[??]] 0.00108 0.00108 0.00108 0.00108
[?.sub.[bar.a]] 0.0251 0.0251 0.0251 0.0251
[?.sub.[??]] 8*e-1 8*e0 8*e1 8*e2
[S.sub.P] 0.00152 0.00187 0.00218 0.00218

Table 9
Different values for the absorber damping while
other parameters are constant (elastic blade)

[µ.sub.[??]] 0.00468 0.00468 0.00468
[?.sub.[bar.a]] 0.0401 0.0401 0.0401
[?.sub.[??]] 4*e-4 4*e-3 4*e-2
[S.sub.C] 0.0112 0.0083 0.0124

[µ.sub.[??]] 0.00468 0.00468 0.00468 0.00468
[?.sub.[bar.a]] 0.0401 0.0401 0.0401 0.0401
[?.sub.[??]] 4*e-1 4*e0 4*e1 4*e2
[S.sub.C] 0.0131 0.0152 0.0161 0.0161
```