Introduction
Computational model and model verification
Computational model of twisted blade with impact and friction
Tab.1 First four natural frequencies under different modal truncations at W=8500 r/min |
N | fn1/Hz | fn2/Hz | fn3/Hz | fn4/Hz | Error for fn1/% | Error for fn2/% | Error for fn3/% | Error for fn4/% | |
---|---|---|---|---|---|---|---|---|---|
4 | 286.0 (282.4) | 1121.2 (1117.0) | 1506.7 (1497.5) | 3975.1 (3958.1) | 1.27 | 0.38 | 0.61 | 0.43 | |
5 | 285.3 (282.4) | 1120.9 (1117.0) | 1502.9 (1497.5) | 3972.6 (3958.1) | 1.03 | 0.35 | 0.36 | 0.37 | |
6 | 284.2 (282.4) | 1119.9 (1117.0) | 1502.0 (1497.5) | 3968.3 (3958.1) | 0.64 | 0.26 | 0.30 | 0.26 | |
7 | 283.6 (282.4) | 1119.9 (1117.0) | 1500.8 (1497.5) | 3969.7 (3958.1) | 0.42 | 0.26 | 0.22 | 0.29 |
Note: Values in () denotes the results obtained from FE model. |
Model verification
Verification of the reduced model
Tab.2 Shrouded blade parameters |
Young’s modulus/GPa | Density/(kg·m−3) | Poisson’s ratio | Disk radius/mm | Blade length/mm | Blade width/mm |
200 | 7800 | 0.3 | 150 | 150 | 40 |
Blade thickness/mm | Stagger angle/(° ) | Shroud length/mm | Shroud width/mm | Shroud thickness/mm | Twist angle/(° ) |
7 | 30 | 40 | 20 | 7 | 10 |
Model verification based on the FE models
Fig.8 Response comparisons among the four models: (a) Impact force on the left side of the shroud, (b) friction force on the left side of the shroud, (c) hysteresis loop on the left side of the shroud, (d) contact state on the left side of the shroud, (e) impact force on the right side of the shroud, (f) friction force on the right side of the shroud, (g) hysteresis loop on the right side of the shroud, (h) contact state on the right side of the shroud, (i) displacements in flexural direction, and (j) displacements in swing direction. |
Tab.3 Calculation times of the four models |
Model | Calculation time/s |
---|---|
Reduced computational model | 5.37 |
Full computational model | 7.66 |
Reduced FE model | 2328.84 |
Full FE model | 3259.67 |
Vibration responses of the shrouded blade under different parameters
Case 1: Effects of initial gap and normal load
Case 2: Effects of coefficient of friction
Fig.12 Amplitude–frequency responses under different D, N0, and m: (a) m = 0.1 under different initial gaps, (b) m= 0.1 under different initial normal preloads, (c) m= 0.3 under different initial gaps, (d) m= 0.3 under different initial normal preloads, (e) m= 0.5 under different initial gaps, and (f) m= 0.5 under different initial normal preloads. |
Fig.13 Resonant response characteristics under different coefficients of friction: (a) Resonant rotational speed under different initial gaps, (b) resonant amplitude under different initial gaps, (c) resonant rotational speed under different initial normal preloads, and (d) resonant amplitude under different normal preloads. |
Case 3: Effects of contact stiffness ratio
Fig.16 Amplitude–frequency responses under different D, N0, and x: (a) x= 0.2 under different initial gaps, (b) x= 0.2 under different initial normal preloads, (c) x = 0.5 under different initial gaps, (d) x = 0.5 under different initial normal preloads, (e) x = 1 under different initial gaps, and (f) x = 1 under different initial normal preloads. |
Fig.17 Resonant response characteristics under different contact stiffness ratios: (a) Resonant rotational speed under different initial gaps, (b) resonant amplitude under different initial gaps, (c) resonant rotational speed under different initial normal preloads, and (d) resonant amplitude under different normal preloads. |