Optimum design of submarine hulls

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1 00 WIT Press, Ashurst Lodge, Southampton, SO0 AA, UK. All rights reserved. ISBN 890 Optimum design of submarine hulls Basem Alzahabi Department of Mechanical Engineering Kettering University, Flint, MI, USA Abstract Vibration characteristics of high performance structure such as submarines are very critical. Submarine hulls are mainly constructed of cylindrical shells. Cylindrical shells are also used in many structural designs, such as offshore structures, liquid storage tanks, and airplane hulls. The vibration characteristics of cylindrical shells present many unique challenges in optimum design of submarine hulls for vibration control. In cylindrical shells, the lowest natural frequency does not necessarily correspond to the lowest wave index. In fact, the natural frequencies do not fall in ascending order of the wave index either. Eigen solutions of cylindrical shells also indicate repeated natural frequencies. These are referred to as double peak frequencies. shapes associated with each natural frequency are combination of (i) Radial (flexural); (ii) Longitudinal (axial); and (iii) Circumferential (torsional) modes. In this paper, uniqueness of modal spectrum, redundancy of modal constraints and nonuniqueness in optimum design of cylindrical shells for vibration requirements are presented. The implications of these characteristics on submarine design are highlighted. Related issues such as the new mode sequence, mode crossing, repeated natural frequencies and stationary modes are also discussed. Introduction Unlike that of simpler structures such as beams and plates, the modal spectrum of cylindrical shell exhibits very unique characteristics. In cylindrical shells, the lowest natural frequency does not necessarily correspond to the lowest wave index shown in Figure. In fact, the natural frequencies do not fall in ascending order of the wave index either as Table indicates.

2 00 WIT Press, Ashurst Lodge, Southampton, SO0 AA, UK. All rights reserved. ISBN HicqhPerjormance 890 Strzlctures and Composites Shell displacements of each mode shape are defined in three orthogonal directions that are associated with radial (flexural), longitudinal (axial), and circumferential (torsional) components. n=l n= n= C?z m = I Circumferential Nodal Pattern. ry... J t m= Longitudinal Nodal Pattern Fig. : Normal mode patterns for simply supported cylindrical shells The coupling between the transverse and inplane vibration is due to the shell curvature. Eigen solutions of cylindrical shells indicate multiple eigenvalues, i.e. repeated natural frequencies with similar mode shapes. These are referred to as double peak frequencies[l]. s shapes associated with membrane shell deformations require a lot of stain energy while mode shapes associated with bending deformation require less strain energy. Realizing that the total potential strain energy in a shell is the sum of both membrane and bending strain energy, the first mode shape corresponding to the lowest total energy may not necessarily correspond to the lowest wave index n. The ratio of membrane strain energy to kinetic energy (or the total strain energy) is high for modes with simple modal patterns n and decrease toward zero as the number of nodal lines increases, while the ratio of the bending strain energy to

3 00 WIT Press, Ashurst Lodge, Southampton, SO0 AA, UK. All rights reserved. ISBN 890 /ii& Per/omance.%wctwesand Composites the to kinetic energy (or the total strain energy) is small for simple nodal patterns and increase with the increase of wave index n []. The natural frequencies that are controlled by the membrane strain energy are approximately independent of the shell thickness change, while the natural frequencies controlled by bending stain energy vary with shell thickness []. Modal dynamics of cylindrical shell A finite element model of the baseline cylindrical shell shown in Figure is analyzed using MSUNASTRAN [] to obtain its modal characteristics. The chosen cylindrical shell represents a segment of submarine hull. It has the following dimensions: length L = 9 in, radius R =l98 in, thickness h = in []. The dynamic characteristics of the cylindrical shell for shear diaphragm boundary conditions are summarized in Table.... >. IL. _ q ,...,...,..., L. " l Length L = 9 in, Radius R = 98 in, Thickness h = in E = 0 x 0psi, v = 0., p =. x IOeIb.sec/in Fig. : Baseline finite element model of simply supported cylindrical shell The results listed in Table lshows the natural frequencies from the finite element analysis and the natural frequencies obtained from an analytical solution derived from references [,]. In developing the analytical solution, an energy approach is used. An energy fhctional which include both bending and membrane strain energies was formulated using the DonnellMushtari formulation of strain displacement. Then RaylighRitz procedure was employed leading to an eigenvalue problem [g].

4 00 WIT Press, Ashurst Lodge, Southampton, SO0 AA, UK. All rights reserved. ISBN High890 Per/omunce S'trzrcturesand Composites No. 8 9 IO Table : Modal Characteristics of baseline cylindrical shell T Frequency (Hz) T FEA Analytic Length = 9 in Ladius R = E=0 x IO psi, v = 0., p =. x IOe Ib.sec/in Total Strain Energy FEA l Analytic l Design optimization A series of thee design optimizations were performed using MSUNASTRAN optimization module (SOL 00) [9]. In each run, a minimum weight optimization is performed for a single modal constraint. Shell thickness is used as a design variable. Nine design variables, each representing a segment thickness, are chosen as shown in Figure. Fig. : Nine design variables shell model In the first run, the modal constraint is imposed on the first natural frequency, The frequency A=/.89 Hz is constraint as follows A'=/#.9 Hz. The modal characteristics of optimized design (Design I)are shown in Table.

5 00 WIT Press, Ashurst Lodge, Southampton, SO0 AA, UK. All rights reserved. ISBN 890 T No. n High I e~ornmncestwctures and Composites Table : Modal characteristics of Design I tn I Baseline Frequency W ) T T n m l Design I Frequency wd Examination of the results listed in Table indicate mode crossing in which mode (n=s, m=l) moved past mode (n=, m=l). Also the natural frequency of mode ( ~ m=l), in the baseline design remained in close proximity to the original natural frequency. In the second run, the modal constraint is imposed on the third natural frequency. The frequency h=. Hz is constraint as follows &l=8 Hz. The modal characteristics of optimized Design ff are shown in Table. TabIe : Modal characteristics of Design ZZ l l Baseline Frequency Design II J No. n m W) n m Frequency (Hz).89 I I I Examination of the results listed in Table indicates the same mode crossing observed in Design. Also the natural frequency of mode ( ~ m=l), in the baseline design remained in close proximity (stationary) to the original natural

6 00 WIT Press, Ashurst Lodge, Southampton, SO0 AA, UK. All rights reserved. 8 ISBN Hij$ 890 Performance.StIuctwes andcomposites frequency. In fact the modal characteristics of Design the Design I. II are almost identical to In the third run, the modal constraint is imposed on the seventh natural frequency. The frequency f~.9hzis constraint as follows jj".9hz. The modal characteristics of optimized Design III are shown in Table. Table : Modal characteristics of Design ZIZ No n m I Baseline Frequency W ) n m l Design III Frequency (Hz) On further exammation of the results listed in Table one can observe the same modal characteristics of Design I and Design II despite the fact that both models were optimized for different modal constraints. Uniqueness of modal spectrum The modal characteristics of the three optimized designs indicate that the natural frequencies of a cylindrical shell are interlinked and uniquely determined based on one of the natural frequency. A summary of the modal characteristics of all optimized designs is listed in Table. One can also observe mode crossing and a stationary natural frequency represented in modes of wave index (w,m=l), and (n=, m=). Attempts to optimize the cylindrical shell for multiple modal constraints would yield no solution except for the case of compatible constraints, i.e. constraints consistent with the modal spectrum as obtained for single modal constraint [ O].

7 00 WIT Press, Ashurst Lodge, Southampton, SO0 AA, UK. All rights reserved. ISBN 890 No T n High Perfi,twmnce Stwctwes md('onzposites 9 Table : Uniqueness of modal spectrum m T Natural Frequency of Optimized Shells (Hz) Design I Nonuniqueness of optimum design Design /I t L Design Ill While the thee optimized designs Design Z, Design U, and Design ZZZ exhibited identical modal spectrum as shown in Table, the values of the design variables (segments' thickness) were not. The values of the design variables of each optimized design are listed in Table. This indicates nonuniqueness of optimal design points. Results in Table show that the symmetry of the design was maintained even though it was not explicitly imposed. Table :Nonuniqueness of optimal design Thickness of Optimized Shells (in) Design I Design /I Design TI =.9 T=.88 T~.8 Tq=. TS=.9 Tgz. T~.8 Ts=.88 T9=.9 Weight =.0 ib TI. T=.09 T =.98 T =.9 TS =. Tg=.9 TT=.98 T8~.09 Tg=. Weight = 8. ib T,=.9 TZz.98 T =.0 T=. TS=.8 T,=. TT =.0 T8~.98 T9=.9 Weight =. lb

8 00 WIT Press, Ashurst Lodge, Southampton, SO0 AA, UK. All rights reserved. 0ISBN High 890 Perjhzance Structures and C onzposites Conclusions Modal characteristics of cylindrical shells are very unique. The natural frequencies are often interlinked resulting in no optimum solution for optimization for incompatible multiple modal constraints. The uniqueness of the modal spectrum causes redundancy in modal constraints. However, there is non uniqueness in optimum design of a cylindrical shell for vibration requirements resulting in different segments thickness. The new modal spectrum shows a new frequency sequence, mode crossing, repeated natural frequencies, and stationary modes. References [l] Soedel, W., Shell Vibration Without Mathematics Part P,SIV, Vol. 9, No., November 9, pp. 0. [] Soedel, W., Shell Vibration Without Mathematics Part II,SIV, Vol. 0, No., April 9, pp.. [] Soedel, W., Similitude Approximation for Vibrating Thin Shells, The Journal of Acoustical Society of America, Vol. 9, No. (part), 9, pp.. [] MXYNASTRAN User s Manual, V., MacNealSchwendler Corporation, L.A., CA, November 99. [] Simites, G.J. and Answani, M., MinimumWeight Design of Stiffened Cylinders Under Hydrostatic Pressure, If Journal of Ship Research, Vol., No., December 9, pp.. [] Leissa, A., Vibration of Shells, NASA SP88, Washington, D.C., 9. [] Markus, S., The Mechanics of Vibrations of Cylindrical Shells, Elsevier Scientific Publishing Company, 988. [8] Alzahabi, B., Redesign of Cylindrical Shells by Large Admissible Perturbations, Ph.D. Thesis, Department of Civil and Environmental Engineering, The University of Michigan, 99. [9] MSUNASTRAN Optimization and Design Sensitivity, V.,MacNeal Schwendler Corporation, L.A., CA, November 99. [lo]alzahabi, B., Bernitsas, M.M., lr Redesign of Cylindrical Shells by Large Admissible Perturbations, Journal of Ship Research, Vol., No., September 00, pp. 8.