Dynamics of a Turbine Blade with an Under-Platform Damper Considering the Bladed Disc’s Rotation
3. Numerical Simulation
(a) time history (b) frequency spectrum
(c) time history (d) frequency spectrum
Figure 6.Steady-state dynamic characteristics of the response (a,b) and dry friction force (c,d).
2. The parameters are taken from Table3. Figures7and8show the simulation results.
Table 3.The parameters of the system.
Parameters Values
m2 0.079 kg
m3 0.07 kg
k2 8×105N/m
F0 600 N
P
y −y
(a) phase diagram (b) hysteretic constructive relation
-3 -2 -1 0 1 2 3
yP 10-3
-2 -1 0 1 2
PV
y
Figure 7.Phase diagram (a) and the hysteretic constructive relationship off1andy1−y2(b) when the system reaches steady-state.
(a) time history (b) frequency spectrum
(c) time history (d) frequency spectrum
Figure 8.Steady-state dynamic characteristics of the response (a,b) and dry friction force (c,d).
The two simulations above are typical among the simulating results in this study. When the blade is at steady-state, the motion is periodic from Figures5a and7a. Figure5b and b show the comparison of hysteretic constructive relation of friction force and relative displacement with and without the Coriolis inertial force, and the two constructive relations in the same figure are obviously different. Considering the bladed disc’s rotation, the Coriolis inertial force exists and changes the normal pressure; therefore, the hysteresis loop is not symmetrical. As the normal pressure is a very critical parameter in the dry friction damper’s design, the dynamic characteristics of the system will be different with that without considering the Coriolis inertial force. In Figures6and8,feis the steady excitation frequency, and only odd multiple frequencies ofy1andf1can be observed. When the mass of the damper, the vibration stiffness of the damper, and the amplitude of external excitation change, there are no fractional frequencies, nor any bifurcation or chaos with the friction contact surface not being separated. The motion of the system is periodic, and the minimum period of the steady-state responseTis equal to that of the external excitation,T=2π/ω0.
3.2. The Decision of Steady-State of the Blade
To supply more reference to dry friction platform damper design in engineering, the blade’s vibrational reduction of the steady-state response and the transient response will be studied in the next section; therefore, it is necessary to get the momentt0when the system reaches steady-state. In this section, a method for deciding the stable state of the system is proposed: combining the normalized cross-correlation function (NCCF) and the bisection method. The principle of this method is as followings: choosing steady-state response of the last period as a reference sequence, and a response before the last period as the target sequence. Window sizehis the length of target sequence which is taken out each time. The correlation coefficient maximumCof reference sequence and target sequence is calculated by the NCCF. The closer that the value ofCis to 1, the more that target sequence is in
agreement with reference sequence. f lagis a given parameter, and whenCis greater than its value the response can be considered steady-state. The step lengthsis the moving length of a target sequence each time, and is changeable via the bisection method, which is used to improve the calculation’s accuracy and efficiency. Comparing target sequence from back to front with the reference sequence, the momentt0can be obtained when the step length equals to 1. A computational scheme of the mothed is shown in Figure9.
BEGIN
The response Sequence
h, j=0, flag, s
Reference sequence
s>1 Move target
sequence s forwards
Choosing target sequence
C C>flag
j=j+1, s=s/2
Move target sequence s backwards
END j>1
s=s/2
yes no
no no
yes yes
Choosing corresponding
reference sequence
NCCF
Figure 9.The computational scheme of the method.
When f lagis 0.999, 0.998, and 0.997, the other parameters are shown in Table2. The results are shown in Figure10. The difference between the two lines has been amplified by three times for clarity.
The momentt0when the system reaches steady-state was obtained. Whenf lagis 0.999,t0satisfies the accuracy requirement.
t
Py
t
t
Figure 10.The results with different value of flag.
Witht0, the response of blade beforet0can be defined asy1t, and aftert0can be can be defined as y1s; therefore,y1is divided intoy1tandy1s.
3.3. The Vibration Reduction Characteristics of the System
Based on the analyses done in Sections3.1and3.2, the influence of damper mass, damper vibration stiffness, and external excitation amplitude on the vibration reduction characteristics of the system are studied in this section. Relevant parameters were set as in Table1. The other parameters are given in the following simulation.
Edsis the vibrational power reduction rate of the steady-state response of the blade. As the steady-state response is periodic,Tis the minimum period of the steady-state response, which is equal to the period of the external excitation with the value of 2π/ω0. Therefore,Edscan be expressed as Equation (8).Edtis the average power-reduction rate of the transient response of the blade, and can be expressed as Equation (9). The relative displacement of the blade without an under-platform damper isy1, and the moment when the system without an under-platform damper reaches steady-state ist0. Similarly, the response of the blade beforet0can be defined asy1t, and aftert0can be defined asy1s. The response is divided intoy1tandy1s.
Eds=
Ty21dt/T−
Ty21dt/T
Ty21dt/T =
Ty21dt−
Ty21dt
Ty21dt =
T(y21−y21)dt
Ty21dt (8)
Edt= t0
0 y21dt/t0−t0 0 y21dt/t0 t0
0 y21dt/t0 = t0
0 y21dt−t0
0 t0y21dt/t0 t0
0 y21dt =
t0
0 (y21−t0y21/t0)dt t0
0 y21dt
(9) The maximum values ofy1tandy1sarey1tmandy1smrespectively, Similarly, the maximum values ofy1tandy1sarey1tmandy1smrespectively.Adsis the reduction rate ofy1smandAdtis the reduction rate ofy1tm.AdtandAdsare expressed in Equation (10).
⎧⎪⎪⎪⎨
⎪⎪⎪⎩ Ads=y1smy−y1sm 1sm
Adt=y1tmy−y1tm 1tm
(10)
3.3.1. The Effect of Damper Mass on the Vibration Reduction
When the working speed is constant, the damper mass has a great influence on the normal pressure. The numerical simulation parameters are taken from Table4.
Table 4.The parameters of the system.
Parameters Values m2 m3+0.009kg m3 0.04 kg~0.08 kg
k2 8×105N/m
F0 400 N
The results are as follows:
In Figure11,EdtandEdsvary with the increase of the damper massm3; some peaks ofEdtandEds are extant. There is a significant reduction of vibrational power with proper damper mass adding to the blade. In Figure12,Adtincreases with the damper mass’s increase, whileAdsfluctuates while the damper mass increases.y1tmandy1smreduce significantly with the proper damper mass adding to the blade. From Figures11and12, the laws ofEdsandAdsvarying withm3are basically the same, while the laws ofEdtandAdtare obviously different, as the transient response of the blade is complicated.
The maximum values ofEdsandAdsare smaller than those ofEdtandAdtwith the same parameters.
Figure 11.The influence of damper mass on the vibrational power reduction.
Figure 12. The influence of damper mass on the reduction of the maximum absolute value of vibrational response.
3.3.2. The Effect of a Damper’s Vibrational Stiffness on the Vibration Reduction
The effect of a damper’s vibration stiffness on the vibration reduction of the blade with an under-platform damper was studied. The parameter values are shown in Table5.
Table 5.The parameters of the system.
Parameters Values
m2 0.059 kg
m3 0.05 kg
k2 6×105N/m~1×106N/m
F0 400 N
The results are as follows:
In Figure13,EdtandEdsfluctuate with the increasing of the damper stiffnessk2, and there is a significant reduction of vibrational power with proper damper stiffnessk2. In Figure14, the damper stiffnessk2has an obvious influence onAdtandAds;y1tmandy1smreduce significantly with the proper damper stiffness. From Figures13and14, the laws ofEdsandEdtvarying withk2,are basically the same, while the laws ofEdsandEdtare obviously different. The maximum values ofEdsandAdsare smaller than those ofEdtandAdtwith the same parameters.
Figure 13.The influence of damper stiffness on the vibrational power reduction.
6 7 8 9 10
k1P 105
28.8 28.9 29 29.1
6 7 8 9 10
k1P 105
0 2 4 6
Figure 14. The influence of damper stiffness on the reduction of maximum absolute value of vibrational response.
3.3.3. The Effect of External Excitation Amplitude on the Vibrational Reduction The parameters are shown in Table6. The results are shown in Figures13and14.
Table 6.The parameters of the system.
Parameters Values
m2 0.059 kg
m3 0.05 kg
k2 8×105N/m
F0 200 N~800 N
In Figure15,EdtandEdsbasically decrease with the increase of external excitation amplitudeF0. In Figure16,AdtandAdsdecrease with increasingF0. From Figures15and16, the increasing external excitation amplitude causes the vibrational reduction effect of the blade to decrease, obviously, and a larger normal pressure would be needed to make the damper work well. The laws ofEdsandEdt varying withk2, are basically the same, as areEdsandEdt. Besides,Edtcould be negative withF0 increasing, which needs to be considered when engineering damper designs.
Figure 15.The influence of external excitation amplitude on the vibrational power reduction.
Figure 16.The influence of external excitation amplitude on the reduction of the maximum absolute value of vibrational response.