Research of AUV Shell Mathematics Model

Xiangzhong Meng, Xiuhua Shi, Xiangdang Du College of Marine, Northwestern Polytechnical University, Xi ’an 710072, China Tel: 86-29-8847 4122 E-mail: mengxz@mail.nwpu.edu.cn Qinglu Hao Department of Aircraft Structure Overhaul, AMECO Beijing, Beijing 100621, China The research is supported by graduate starting seed foundation of Northwestern Polytechnical University. No. Z200510 Abstract The finite element analysis and research on the vibration mod e of a certain AUV with the analytic method are expatiate d. Based on the basic equations of thin shells theory, this pap er nalyses and sets up the cylindrical shell vibration mathematical model, and validated the correctness of the m odel by ANSYS. The method presented is ef fective in analyse and dynamical design of AUV weaken vibration and low noise structure.


The balance differential equations of free vibration
A middle surface patch of cylindrical shell and internal forces in the orthogonal coordinate system are shown in Figure 1.α and β are the lines of main curvature, and γ is the normal pointed to convex direction.
Q are the internal forces acted on the α plane and the β plane, 1 k and 2 k are the main curvatures on the α direction and the β direction, R is the radius of the middle surface, and 1 0 k = , 2 k R = , A and B are the Lame coefficients on the α direction and the β direction, and p , 2 p , 3 p are the component of loads on the α direction, the β direction and the γ direction.The plus directions of the internal forces are shown in Figure 2.
The sum of all the internal forces components on the α direction divided by

∑
. The moment of all the internal forces to the α axis is 0, that is

∑
. After operated, the balance equations of the cylindrical shell are: From the last two equations of equations (1.1), the expressions of 1 Q and 2 Q can be reason out.The expressions reckoned in the former three equations and the affects of the cross shearing force 2 Q to the balance on the α direction and the β direction are neglected, then At the state of free vibration, according to D'alembert's Principle, the free vibration balance differential equations of the cylindrical shell are: Where, m is the unit area mass of the cylindrical shell middle surface, u , v and w are the component of displacements on the α direction, the β direction and the γ direction of any point on the middle surface.
The geometric equations (1.4) and the physical equations (1.5) of the cylindrical shell are: Where, 1 ε , 2 ε , 12 ε are the inplane strains on the middle surface, 1 χ , 2 χ , 12 χ are the bending strains on the middle surface E is the elastic modulus, µ is the Poisson's ratio, D is the bending strength, and .
The strains are eliminated by combining the equations (1.4) and the equations (1.5), the result are brought into the equations (1.5), then, the free vibration balance differential equations of the cylindrical shell expressed by displacements are: Where, the differential operator is .

The solution of free vibration eigenfrequency
The solution of balance differential equations of free vibration can be solved by mixed method.Supposed circumferential load, that is 1 2 0 p p = = , the internal force function ( , ) ϕ ϕ α β = is brought in, and supposed, The equations (2.1) and the equations (1.5) are brought in the third equation of the equations (1.2).The normal shell vibrates freely by the force of inertia.The normal free vibration balance differential equations of the cylindrical shell, , the equations (2.2) evolved to: Both of the two ends are supported, and the boundary conditions are, Supposed the expressions of the internal force function ϕ and the deflection w are, sin sin sin (2.5) sin sin sin The boundary conditions (2.4) are fulfiled in the equations (2.5), and the equations (2.5) are brought in the equations (2.3), If the coefficient determinants of ab A and ab B are zero, ϕ and w are not identical to zero, that is, , and supposed , and 3 2 12( 1) The minimum value of b µ is, The minimum eigenfrequency can be computed by 1 λ and 1 µ , that is,

The analysis of a certain AUV shell and emulation by ANSYS
The shell of AUV is made up of spherical shell, cylindrical shell, conical shells and other revolutionary shells by thread coupling, bolt coupling, wedge coupling and hoop coupling.All of them are rigid coupling.Emulation analysis of the cylindrical shell is operated using the ANSYS, and the minimum eigenfrequency is 84.77Hz.

Conclusions
Sum up, the error between the computational result and the emulational result of the minimum eigenfrequency is 1.53%.The different eigenfrequency can be calculated by different a λ and b µ , the error between the computational result and the emulational result of the eigenfrequency is no more than 5%.The method is effective in analyse and dynamical design of AUV weaken vibration and low noise structure.