This paper proposes a new non-intrusive trigonometric polynomial approximation interval method for the dynamic response analysis of nonlinear systems with uncertain-but-bounded parameters and/or initial conditions. This method provides tighter solution ranges compared to the existing approximation interval methods. We consider trigonometric approximation polynomials of three types: both cosine and sine functions, the sine function, and the cosine function. Thus, special interval arithmetic for trigonometric function without overestimation can be used to obtain interval results. The interval method using trigonometric approximation polynomials with a cosine functional form exhibits better performance than the existing Taylor interval method and Chebyshev interval method. Finally, two typical numerical examples with nonlinearity are applied to demonstrate the effectiveness of the proposed method.
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Figure 1
(Color online) Errors of (2+arctan
Figure 2
(Color online) Two-degree-of-freedom quarter-car model of the vehicle.
Figure 3
(Color online) Comparison of the sprung mass displacement calculated using the trigonometric methods in the three forms and the scanning method.
Figure 4
(Color online) Comparison of the unsprung mass displacement calculated using the trigonometric methods in the three forms and the scanning method.
Figure 5
(Color online) Comparison of the sprung mass displacement calculated using the Chebyshev method, the TPCIAT method and the scanning method.
Figure 6
(Color online) Comparison of the unsprung mass displacement calculated using the Chebyshev method, the TPCIAT method and the scanning method.
Figure 7
(Color online) Comparison of the top pendulum angle calculated using the trigonometric methods in the three forms and the scanning method.
Figure 8
(Color online) Comparison of the bottom pendulum angle calculated using the trigonometric methods in the three forms and the scanning method.
Figure 9
(Color online) Comparison of the top pendulum angle calculated using the Chebyshev method, the TPCIAT method and the scanning method.
Figure 10
(Color online) Comparison of the bottom pendulum angle calculated using the Chebyshev method, the TPCIAT method and the scanning method.
Parameters |
|||||||
Mean values |
375 |
60 |
1000 |
15000 |
200000 |
1.5×106 |
2×107 |
Intervals |
? |
? |
[900,1100] |
[135,165]×102 |
? |
? |
Methods |
Scanning |
TPCSIAT |
TPSIAT |
TPCIAT |
Chebyshev |
Calculation time (s) |
643.5 |
143.4 |
15.4 |
30.4 |
30.9 |
Methods |
Scanning |
TPCSIAT |
TPSIAT |
TPCIAT |
Chebyshev |
Calculation time (s) |
108.1 |
42.0 |
13.5 |
17.4 |
17.5 |
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