PROBABILISTIC MODELLING OF DYNAMIC RESPONSE OF UNDERWATER VEHICLE STRUCTURE VIA MAXIMUM ENTROPY METHOD

doi:10.6052/0459-1879-17-022

Abstract

Abstract:

Hydrodynamic loads of underwater vehicle structure have a high degree of randomness in time and space, which leads to random responses of structure such as maximum internal forces, time and position they taking place, etc. Dynamic response of underwater vehicle will also be nonlinear due to different tension and compression stiffness of its connecting or separated structures. To analyse the influence of nonlinearity of connection on statistics of response of underwater vehicle based on limited sample data of hydrodynamic external load, probabilistic modelling of dynamic response of underwater vehicle is completed by using maximum entropy method in which statistical moments of response samples are obtained from a simplified dynamic model of the structure. Probability density functions of maximum internal forces, time and position they taking place, are proposed separately to predict the most dangerous cases of the underwater vehicle. Accuracy of results proposed here is verified through Monte Carlo simulation. Effects of nonlinear parameters of connections on dynamic response are studied and discussed. Final conclusions are as follows: Nonlinearity of connection leads to axial force response of structures under transverse loads; peak of probabilistic density function of maximum axial force of the structure increases with nonlinear degree of connections which leads to significant effect of propagation of uncertainty; internal force response of structure with nonlinear connections under normal random loads is non-normal; position of maximum internal force response of underwater vehicle structure is influenced by degree of nonlinearity of connections. These findings can provide technical support for structural optimization.

Key words: maximum entropy method, dynamic response, probabilistic modelling, external loads with randomness in time and space, Monte Carlo simulation, underwater vehicle

CLC Number:

O324

Zhou Chunxiao, Wang Ruiqiong, Nie Zhaokun, Li Gang, Zeng Yan. PROBABILISTIC MODELLING OF DYNAMIC RESPONSE OF UNDERWATER VEHICLE STRUCTURE VIA MAXIMUM ENTROPY METHOD[J]. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(1): 114-123.

Figures/Tables 23

Fig. 1 Three section linear beams with nonlinear connection structures for underwater vehicle structure

Fig. 2 Distribution of pressure difference on surface of a section

Fig. 3 Schematic diagram of external loads

Table 1 Section position

Section	1	2	3	4	5	6	7	8
Position/m	0.9	1.6	2.3	3.0	3.6	4.2	4.9	5.6

Fig. 4 Diagram of force acting on a beam unit

Fig. 5 Figures of space-time distribution of axial force

Fig. 6 Figures of space-time distribution of bending moment

Table 2 The first four moments of the maximum internal force of the beam with nonlinear connections

	$E(X^{1})$	$E(X^{2)}$	$E(X^{3)}$	$E(X^{4)}$
$F$/kN&	78.01	9.27	0.13	2.47
$t_{F}$/ms	100.63	23.37	2.17	5.88
$L_{F}$/m	7.27	0.21	$- 6.57$	64.52
$M$/(kN$\cdot$m)	188.52	14.98	$- 0.16$	2.80
$t_{M}$/ms	123.46	2.21	$- 0.37$	2.99
$L_{M}$/m	7.20	0.32	1.03	6.71

Table 2 The first four moments of the maximum internal force of the beam with nonlinear connections

	$E(X^{1})$	$E(X^{2)}$	$E(X^{3)}$	$E(X^{4)}$
$F$/kN&	78.01	9.27	0.13	2.47
$t_{F}$/ms	100.63	23.37	2.17	5.88
$L_{F}$/m	7.27	0.21	$- 6.57$	64.52
$M$/(kN$\cdot$m)	188.52	14.98	$- 0.16$	2.80
$t_{M}$/ms	123.46	2.21	$- 0.37$	2.99
$L_{M}$/m	7.20	0.32	1.03	6.71

Fig. 7 Probability density curve of maximum axial force

Fig. 8 Probability density curve of the time of occurrence of maximum axial force

Fig. 9 Probability density curve of the location of maximum axial force

Fig. 10 Probability density curve of maximum bending moment

Fig. 11 Probability density curve of the time of occurrence of maximum bending moment

Fig. 12 Probability density curve of the location of maximum bending moment

Table 3 The first four moments of the maximum internal force on dangerous location of the beam with nonlinear connections

	$E(X^{1})$	$E(X^{2})$	$E(X^{3})$	$E(X^{4})$
$F_{7.3m}$/kN	73.12	8.67	0.05	2.52
$M_{7.1m}$/(kN$\cdot$m)	184.15	14.14	$- 0.18$	2.84

Table 3 The first four moments of the maximum internal force on dangerous location of the beam with nonlinear connections

	$E(X^{1})$	$E(X^{2})$	$E(X^{3})$	$E(X^{4})$
$F_{7.3m}$/kN	73.12	8.67	0.05	2.52
$M_{7.1m}$/(kN$\cdot$m)	184.15	14.14	$- 0.18$	2.84

Fig. 13 Probability density curve of maximum axial force at 7.3 m

Fig.14 Probability density curve of maximum bending moment at 7.1 m

Fig. 15 6 cases with different stiffness curves and values of ratio of tension and compression for nonlinear connections

Fig. 16 Probability density curve of maximum axial force

Fig. 17 Probability density curve of maximum bending moment

Table 4 Probabilities of dangerous bending moments for 6 cases

Case	1	2	3	4	5	6
$P$ /%	0.07	0.89	1.59	2.93	2.52	1.68

Table 4 Probabilities of dangerous bending moments for 6 cases

Case	1	2	3	4	5	6
$P$ /%	0.07	0.89	1.59	2.93	2.52	1.68

Fig. 18 Probability density curve of the time of maximum bending moment

Fig. 19 Probability density curve of the location of maximum bending moment

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