Dynamic modeling of turbine blade with friction damper and study of the resonant response of blade.
Service failure of gas turbine blades in many instances can be attributed to high cycle fatigue caused by large resonant amplitudes. In order to avoid such failures, designers of aircraft engines frequently incorporate friction devices into turbine designs to increase damping and reduce vibratory stresses. These devices are frequently of the type designated herein as "blade-to-ground" dampers. A blade -to-ground damper provides a link between a vibrating point on the blade (see fig.1) and a relatively rigid structure such as the cover plate. The damper transmits a load through a friction contact which dissipates energy when slip occurs.
[FIGURE 1 OMITTED]
2. LITERATURE REVIEW
For a number of years, considerable effort has been made towards understanding the effects of friction on the dynamic response of structures. But there is a lack of theoretical model that can predict the response of the experiments.
[FIGURE 2 OMITTED]
The response of turbine blade under the influence of friction damping is studied in different ways by different researchers (Berger and Krousgrill 2002, Cameron and Griffin 1989, 1990, Csaba 1991, Den Hartog 1931, Ferri and Dowell 1988, Griffin 1980, 1990, Hundal 1979, Johansson 1992, Menq et al 1986, 1986, Metherell and Diller 1968, Mindlin et al 1951 and Salinturk et al 2001). But, there is a need for a good mathematical model to simulate the parameters affecting the response of the turbine blade easily. To address the lack of research on this issue, a mathematical model is developed for the combined blade with damper in the present work to study the system response in a effective way.
The equations 1, 2 3 4 and 5 were collected from Venkataramaiah (2006) to determine the stiffness of damper ([K.sub.2]) and damping coefficient ([C.sub.eq]).
Normal load acting on the damper, N = [[beta].sub.1]L + 2/3[[beta].sub.2]L (1)
The normalised energy dissipation due to friction,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)
Energy dissipation, [E.sub.d] = [E.sub.dn]EAL (3)
The equivalent viscus damping coefficient, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)
Damper stiffness, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)
3. DEVELOPMENT OF COMBINED BLADE WITH DAMPER SYSTEM
The combined system is modeled by considering the turbine blade as a cantilever beam which is fixed at one end and a damper is attached to the blade as shown in Fig.1. A dynamic model is developed for the combined blade and damper system as shown in Fig.3. This dynamic model consists of the elements such as modal mass of blade (m), modal stiffness of blade ([K.sub.1]), stiffness of damper ([K.sub.2]) and damping coefficient ([C.sub.eq]). This model has the ability to predict the response of structure.
Mass is excited by harmonic force ([P.sub.1]). Displacement of mass is [x.sub.1] and damper displacement is [x.sub.2] produced due to excitation. In the analysis, governing equations are developed for the model as shown in 6,7.
[FIGURE 3 OMITTED]
Assumptions of model
1. Excitation force is harmonic in nature.
2. Friction damping is replaced by equivalent viscous damping in model.
3. Force at damper end is equal to excitation force
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)
Solution of above equations is easily found by using complex algebra procedure by considering
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)
From eq.6, 7, and 8 the solution of equation as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)
To simplify analysis, make the above equation as non-dimensional, by considering
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)
Maximum amplitude of excitation at [[bar.[omega].sub.1] = [[bar].[omega].sub.n] is
[X.sub.1] = M.F ([P.sub.o]/[K.sub.1] (12)
4. NUMERICAL SIMULATIONS AND RESULTS
Here MATLAB code is developed to perform various numerical simulations for the parameters which affects the response of turbine blade under the influence of friction damping. The data are considered for simulations as follows:
Elasticity of damper material (E) =2x[10.sup.5] N/[mm.sup.2]
Length of damper (L)=25 mm
Area of cross section of damper(A)=16 [mm.sup.2]
Coefficient of friction ([mu])=0.12 - 0.4
Max. Applied force at damper end (Pa)=800 N
Modal mass of blade (m)= 0.1243 Kg
Modal stiffness of blade ([K.sub.1])= 9.3743X[10.sup.5] N/m
Natural frequency of blade ([f.sub.n])=437 Hz
Circular natural frequency of blade ([[bar.[omega].sub.n])=2700 rad/sec
Circular frequency of excitation ([[bar].[omega].sub.1])=[[bar].[omega].sub.n]
Elasticity of blade material ([E.sub.b])=2.1X[10.sup.5] N/[mm.sup.2]
Density of blade material ([rho])=8.53X[10.sup.-6] N/[mm.sup.3]
Cross section of blade at root (bXt)=40X2.5 [mm.sup.2]
Blade and damper material=1080 cold rolled steel
4.1. Effect of excitation force on resonant amplitude of blade vibration
Fig.4. shows that the resonant amplitude of blade is changed with increase of normal load at different excitation levels. The amplitude of vibration is decreased with increase of normal load upto 200 N and after this, the amplitude is again increased. This is due to the increase of damper stiffness with the increase of normal load after 200 N.
Coefficient of friction=0.17
Excitation force for
Curve-1 =15 N Curve-2 =18 N Curve-3 =21 N
[FIGURE 4 OMITTED]
4.2. Effect of coefficient of friction on amplitude of blade vibration
[FIGURE 5 OMITTED]
Fig.4. shows that the resonant amplitude of blade is changed with increase of normal load at different coefficient of friction. The position of curves changed with change of coefficient of friction and optimum normal load is shifted. This depends upon the amount of excitation force and damper stiffness.
Coefficient friction for Curve-1 =0.17, cuve-2 =0.23, cuve-3=0.29, curve-4=0.35, curve-5 =0.41 Excitation force ([P.sub.o]) =20 N
Numerical simulations are performed for the parameters which affect the resonant response of blade to analyze the system. Dynamic model of blade and damper system which is developed here is effectively reflects the performance of damper. This model would optimize the friction damper and is capable of simulating response parameters under all conditions of damper. The parametric studies show that it is possible to draw qualitative conclusions using the blade with damper model developed in this work, and it would reduce the laboratory work
NOTATIONS A Damper cross-section area in [mm.sup.2] N Total normal acting on damper in Newton E Modulus of elasticity of damper material in N/[mm.sup.2] L Length of damper in mm [U.sub.a] Max. Displacement of damper end in mm [mu] Coefficient of friction [[beta].sub.1], Normal load coefficients in N/mm [[beta].sub.2] [[alpha].sub.a] Maximum slip length in mm [[alpha].sub.an] Slip coefficient [P.sub.a] Max force at damper end in Newton [K.sub.2] Stiffness of damper N/m m Modal mass of blade in Kg [K.sub.1] Modal stiffness of blade material in N/m [[bar.[omega].sub.1] Circular frequency of excitation in rad/sec [[bar.[omega].sub.2] Circular natural frequency of blade in rad/sec [xi] Damping factor [C.sub.eq] Equivalent viscous damping coefficient in N-sec/m [X.sub.1] Max. Amplitude at tip of blade in mm [X.sub.2] Displacement of the damper in mm. [P.sub.o] Excitation force in N.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
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* P. Venkataramaiah and ** G. Krishnaiah
* Department of Mechanical Engineering, Srikalahasteeswara Institute of Technology, SRIKALAHASTI- 517644, (A.P.), INDIA.