Nonlinear Dynamic Behaviour Analysis of a Clutch System with Uncertainties Using Polynomial Chaos and the Constrained Harmonic Balance Method
Table 4
Comparison of the limit cycles of a clutch system with several uncertain parameters between the MC method and the combination of the CHBM and ngPC: three different values of the friction coefficient and with different dispersions interval for uncertain parameters.
N
Number of uncertain parameters r
1
2
5
8
μ
μ, kp
μ, kp, kf, γ1, γ2,
μ, kp, kf, γ1, γ2, r1, r2, l
Number of resolutions for the system of equations (see (13))
3
9
42
90
0.01
0.05
0.01
0.05
0.01
0.05
0.01
0.05
μ = 0.25
Maximum amplitude of displacements (10e − 4, rad)
216
216
216
220
218
254
218
240
Relative error of the maximum amplitude of displacements (%)
7.81
7.81
7.72
7.13
7.64
8.16
8.07
37.7
Minimum amplitude of displacements (10e −, rad)
204
204
204
198
200
180
198
168
Relative error of the minimum amplitude of displacements (%)
9.93
9.93
9.96
10.7
10.1
12.3
10.4
19.4
Maximum amplitude of velocities (rad·s−1)
350
350
352
360
354
398
362
410
Relative error of the maximum amplitude of velocities (%)
8.14
8.14
8.05
7.47
7.98
8.96
8.41
24.9
Minimum amplitude of velocities (rad·s−1)
330
330
330
320
324
286
320
274
Relative error of the minimum amplitude of velocities (%)
10.2
10.2
10.2
11.0
10.4
12.5
10.5
14.7
μ = 0.35
Maximum amplitude of displacements (10e − 4, rad)
256
256
256
262
258
276
258
282
Relative error of the maximum amplitude of displacements (%)
3.73
3.73
4.02
3.44
2.89
3.18
3.87
3.35
Minimum amplitude of displacements (10e − 4, rad)
244
244
244
238
240
224
236
212
Relative error of the minimum amplitude of displacements (%)
4.61
4.61
4.63
4.93
4.59
5.36
5.97
6.89
Maximum amplitude of velocities (rad·s−1)
416
416
418
426
420
454
428
494
Relative error of the maximum amplitude of velocities (%)
4.12
4.12
4.09
3.85
3.63
3.79
4.27
4.11
Minimum amplitude of velocities (rad·s−1)
396
396
396
384
388
356
386
338
Relative error of the minimum amplitude of velocities (%)
4.98
4.98
4.97
5.30
5.21
5.70
5.06
5.19
μ = 0.5
Maximum amplitude of displacements (10e − 4, rad)
304
304
304
310
306
322
306
330
Relative error of the maximum amplitude of displacements (%)
1.74
1.74
1.72
1.61
1.73
1.51
1.79
1.58
Minimum amplitude of displacements (10e − 4, rad)
290
290
290
284
290
272
286
260
Relative error of the minimum amplitude of displacements (%)
2.16
2.16
1.90
2.30
1.07
2.41
2.64
3.07
Maximum amplitude of velocities (rad·s−1)
494
494
496
506
496
528
508
574
Relative error of the maximum amplitude of velocities (%)
2.26
2.26
2.24
2.14
2.26
2.18
2.24
2.41
Minimum amplitude of velocities (rad·s−1)
470
470
470
460
470
434
460
410
Relative error of the minimum amplitude of velocities (%)
