Modal vibration control of submarine hulls

Modal vibration control of submarine hulls B. Alzahabi Department of Mechanical Engineering, Kettering University, USA Abstract Cylindrical shells are widely used in many structural designs, such as offshore structures, liquid storage tanks, submarine hulls, and airplane hulls. Most of these structures are required to operate in a dynamic environment. The acoustic signature of submarines is very critical in such a high performance structure. Submarines are not only required to sustain very high dynamic loadings at all times, but must also be able to maneuver and perform their functions under the sea without being detected by sonar systems. Reduction of sound radiation is most efficiently achieved at the design stage, and the acoustic signatures may be determined by considering operational scenarios and modal characteristics. The acoustic signature of submarines is generally of two categories: broadband which has a continuous spectrum; and a tonal noise which has discrete frequencies. Therefore, investigating the dynamic characteristics of a submarine hull is very critical in developing a strategy for modal vibration control for specific operating conditions. During the design optimization of a submarine hull, one is faced with some unique challenges. Unlike that of simpler structures such as beams and plates, the modal spectrum of a cylindrical shell exhibits very unique modal characteristics. The interrelationship between modes usually results in mode crossing, uniqueness of the modal spectrum, and the redundancy of modal constraints. Design optimization due to modal frequency constraints also results in nonunique solutions. Those designs must be examined for their modal frequency response to determine the best suitable design. In this paper, a strategy for modal vibration control is investigated. First, the modal characteristics of a submarine hull are examined. Second, the optimum design for modal frequency constraints is established. The frequency responses of the resulting optimum designs are compared. Third, a frequency response optimization is presented and compared with other models. 2004 WIT Press, www.witpress.com, ISBN -8532-77-5

56 High Performance Structures and Materials II Modal characteristics of submarine hull The modal spectrum of cylindrical shell exhibits very unique characteristics. Unlike that of simpler structures such as beams and plates, the lowest natural frequency in cylindrical shells 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 indicated in Table. The eigen solutions also indicate multiple eigenvalues, i.e. repeated natural frequencies with similar mode shapes. These are referred to as double peak frequencies []. 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 = n = 2 n=3 Circumferential Nodal Pattern m = m = 2 Longitudinal Nodal Pattern Figure : Normal mode patterns for simply supported cylindrical shells. The shell curvature results in a coupling between the transverse and in-plane vibration. Modes shapes associated with membrane shell deformations require a lot of strain energy while mode shapes associated with bending deformation require less strain energy. Since the total potential strain energy in a shell is the summation of both membrane and bending strain energy, the first mode shape corresponds to the lowest total energy that might not necessarily be at the lowest wave index n. The ratio between the membrane strain energy and the kinetic energy (or the total strain energy) is high for modes with simple modal patterns n 2004 WIT Press, www.witpress.com, ISBN -8532-77-5

High Performance Structures and Materials II 57 and decrease toward zero as the number of nodal lines increases, while the ratio of the bending strain energy to the to kinetic energy (or the total strain energy) is small for simple nodal patterns and increase with the increase of wave index n [2]. 24000 Total Strain energy Membrane Strain Energy Bending Strain Energy Energy Factor (lb.in) 9000 4000 9000 4000-000 2 3 4 5 6 7 8 'n' - Number of Circumferential Waves in the mode shape for m=. Figure 2: Energy distribution. Natural frequencies dominated by the membrane strain energy are approximately independent of the shell thickness change [3], and remain unchanged during shell optimization [4], while the natural frequencies controlled by bending stain energy vary with shell thickness. 2 Modal characteristics The finite element model of the cylindrical segment of a submarine hull, shown in Figure 3 is analyzed using the finite element software MSC.NASTRAN [5] to obtain its modal characteristics. The cylindrical shell segment has the following dimensions: length L = 594 in, radius R =98 in, thickness h = 2 in. The material properties of the shell are: Young s modulus E = 30 x 0 6 psi, Poisson s ratio ν = 0.3, Density ρ = 7.324 x 0-4 lb.sec 2 /in 4. The modal characteristics of the cylindrical shell for shear diaphragm boundary conditions are summarized in Table. 2004 WIT Press, www.witpress.com, ISBN -8532-77-5

58 High Performance Structures and Materials II Figure 3: Table : Baseline finite element model of submarine hull segment. Modal Characteristics of baseline submarine hull segment. Mode Mode Frequency Strain Energy Density (lb/in 2 ) No. n m (Hz) Membrane Bending 2 3 4 5 6 7 8 9 0 2 3 4 4 4 5 5 3 3 6 6 7 7 8 6 6 5 2 2 2 2.89 2.89 4.3 4.3 7.39 7.39 8.39 8.39 23.54 23.54 27.07 27.93 27.93 28.74 2086.48 2086.50 945.20 945.20 567.23 567.23 52.98 52.99 34.74 34.74 295.29 7265.06 7265.7 2564.43 95.40 95.40 2997.83 2997.83 354.23 354.23 664.88 664.88 0593.64 0593.64 472.68 836.35 836.33 3742.67 3 Design optimization I Modal frequency constraints A series of design optimizations were performed using SOL 200 module of MSC.NASTRAN [6]. For the optimization, the uniform submarine hull is segmented in the finite element model as a segmented cylinder made of nine equal length cylindrical segments along its length as shown in figure 4. 2004 WIT Press, www.witpress.com, ISBN -8532-77-5

High Performance Structures and Materials II 59 The thickness of each segment is taken as a design variable, totaling nine design variables for the optimization. Figure 4: Table 2: Nine design variables of baseline model. Non-uniqueness of design optimization. Thickness Baseline Design I Design II Design III Design IV T 2.000 2.52 4.076 2.523 2.984 T 2 2.000 2.583 4.20 2.805 2.898 T 3 2.000 2.888 2.737 2.652 2.709 T 4 2.000 2.823.834 2.839 2.647 T 5 2.000 2.799.994 2.94 2.670 T 6 2.000 2.823.834 2.839 2.647 T 7 2.000 2.888 2.737 2.652 2.709 T 8 2.000 2.583 4.20 2.805 2.898 T 9 2.000 2.52 4.076 2.523 2.984 Mass (lb.sec 2 /in) 082.9 469.7 656. 477. 52.8 Four different designs were obtained while optimizing each segment thickness to attain desired natural frequencies [7] for modes (n=4 to 7, m=). Predetermined natural frequencies for the four modes were used as design objectives in this optimization program utilizing normal mode analysis. The optimized designs were named as Design I, II, III & IV. Design I: The design objective is to increase the second mode (n = 4, m = ) natural frequency from baseline frequency f =2.89 Hz to design frequency f '=4.92 Hz. Design II: The design objective is to increase the third mode (n = 5, m = ) natural frequency from baseline frequency f 2 =4.3 Hz to design frequency f 2 '=8.69 Hz. 2004 WIT Press, www.witpress.com, ISBN -8532-77-5

520 High Performance Structures and Materials II Design III: The design objective is to increase the fifth mode (n = 6, m = ) natural frequency from baseline frequency f 3 =8.39 Hz to design frequency f 3 '=24. 94 Hz. Design IV: The design objective is to increase the sixth mode (n = 5, m = ) natural frequency from baseline frequency f 4 =23.54 Hz to design frequency f 4 '=3.66 Hz. The optimization results show non-unique solution, i.e. there are many very different solutions having almost the same value of the goal function. The optimized thickness for the above four designs are as shown in Table 2. While the optimization process results in four non-unique solutions, they exhibit unique modal spectrum as listed in Table 3. Table 3: Uniqueness of modal spectrum. Mode No. n M Baseline n m Design I Design II Design III Design IV 4 2.89 4 4.90 5.3 4.9 4.79 2 4 2.89 4 4.90 5.3 4.9 4.79 3 5 4.3 3 7.75 8.07 7.77 7.93 4 5 4.3 3 7.75 8.07 7.77 7.93 5 3 7.39 5 8.46 8.67 8.45 8.5 6 3 7.39 5 8.46 8.67 8.45 8.5 7 6 8.39 6 24.90 25.3 24.89 24.63 8 6 8.39 6 24.90 25.3 24.89 24.63 9 7 23.54 2 30.86 32.45 30.94 3.60 0 7 23.54 2 30.86 32.45 30.94 3.60 8 27.07 5 2 3.43 32.8 3.42 3.6 2 6 2 27.93 5 2 3.43 32.8 3.42 3.6 3 6 2 27.93 7 3.74 33.69 3.82 3.98 4 5 2 28.74 7 3.74 33.69 3.82 3.98 4 Modal frequency responses analysis The non-uniqueness of the optimization solution raises the question of choice of an appropriate design. To find the best solution among the various designs an additional analysis will be performed, i.e. the frequency response of the different designs. The steady state response of the shell due to a sinusoidal input force at discrete frequencies is studied using the modal frequency response analysis [5] in MSC.NASTRAN. Unit sinusoidal force acting in the X-direction is applied at the center of the cylinder while rigid bar is used to connect the center to a point on the cylinder as shown in figure 4. A sine sweep over the frequency range of 2 Hz to 32 Hz is performed and the structure s response (X-direction displacement) at the point of response is 2004 WIT Press, www.witpress.com, ISBN -8532-77-5

High Performance Structures and Materials II 52 examined. Since, the structure is considered linear in the frequency response analysis; the response to the sinusoidal excitation will also be sinusoidal response, vibrating with the same frequency as the input but at a different phase. Figure 5, shows the cylinder s displacement in the X direction (magnitude in logarithmic scale) at the response point for the all four designs. Table 4, shows the values of the RMS (Root Mean Squares) summation of X-direction displacement for the various frequencies at the response point. Figure 5: Cylinder excited by a point force..e-03 Design I Design II Design III Design IV Baseline.E-04.E-05 X displ (log scale).e-06.e-07.e-08.e-09.e-0 2.0 4.0 6.0 8.0 20.0 22.0 24.0 26.0 28.0 30.0 32.0 Frequency (Hz) Figure 6: Modal frequency response for all designs. 2004 WIT Press, www.witpress.com, ISBN -8532-77-5

522 High Performance Structures and Materials II Table 4: Design summary optimization I. Baseline Design I Design II Design III Design IV Σ RMS 3.7E-07.E-06 5.4E-08.2E-07 7.6E-08 5 Optimization II - modal frequency response approach To further investigate an appropriate optimized design for the submarine hull segment, a frequency response optimization is performed. The RMS was taken as the objective function and considering a desired frequency as a design constraint. This was accomplished using the multidisciplinary design optimization module in MSC.NASTRAN. For each of the modal frequency response optimization, unit sinusoidal force in the X-direction is applied on a circular ring of nodes lying at the start of the fifth segment. The forces applied are in phase, swept through discrete frequencies from 2 Hz to 32 Hz. Subjecting the cylindrical shell to excitation forces all along the circular ring ensures all mode shapes are excited irrespective of the excitation location. Also to take into account the responses due to all modes, the Root Mean Squares (RMS) of the X- direction displacement at each of the excitation locations is summed up for every discrete frequency of excitation used. During optimization, the design objective is designed to reduce this RMS summation value. The design constraint in each design case is the predetermined frequency values for the first four mode shapes. The resulting four design variants are named as Design V, VI, VII and VIII. Table 5: Non-uniqueness of design optimization II. Thickness Baseline Design V Design VI Design VII Design VIII T 2.000 2.655 4.23 2.523 2.978 T 2 2.000 2.66 4.82 2.804 3.035 T 3 2.000 2.82 2.673 2.664 2.666 T 4 2.000 2.874.775 2.836 2.655 T 5 2.000 2.764.878 2.905 2.626 T 6 2.000 2.874.775 2.836 2.655 T 7 2.000 2.82 2.673 2.664 2.666 T 8 2.000 2.66 4.82 2.804 3.035 T 9 2.000 2.655 4.23 2.523 2.978 Mass (lb.sec 2 /in) 082.9 490. 660.4 477.5 52.7 Design V: The objective is to minimize the root mean square sum of X-dir displacements and the design constraint is that the natural frequency of the second mode (n = 4, m = ) is f '=4.92 Hz. 2004 WIT Press, www.witpress.com, ISBN -8532-77-5

High Performance Structures and Materials II 523 Design VI: The objective is to minimize the root mean square sum of X-dir displacements and the design constraint is that the natural frequency of the third mode (n = 5, m = ) is f 2 '=8.69 Hz. Design VII: The objective is to minimize the root mean square sum of X-dir displacements and the design constraint is that the natural frequency of the fifth mode (n = 6, m = ) is f 3 '=24.94 Hz. Design VIII: The objective is to minimize the root mean square sum of X-dir displacements and the design constraint is that the natural frequency of the sixth mode (n = 7, m = ) is f 4 '=3.66 Hz. Optimized segment thickness for the above designs is shown in Table 5. The corresponding modal spectrums of these four designs are shown in Table 6. The corresponding modal spectra of these four designs are shown in Table 6. The modal frequency functions at the response point in the X direction displacement are shown in logarithmic scale in Figure 7. The corresponding values of the RMS (Root Mean Squares) summation of X- direction displacement for the all frequencies at the response point are listed in Table 7. Table 6: Uniqueness of modal spectrum. Mode No. n m Baseline n m Design V Design VI Design VII Design VIII 4 2.89 4 4.94 5.6 4.90 4.8 2 4 2.89 4 4.94 5.6 4.90 4.8 3 5 4.3 3 7.8 8.06 7.78 7.94 4 5 4.3 3 7.8 8.06 7.78 7.94 5 3 7.39 5 8.52 8.68 8.45 8.8 6 3 7.39 5 8.52 8.68 8.45 8.8 7 6 8.39 6 25.02 25.22 24.89 24.69 8 6 8.39 6 25.02 25.22 24.89 24.69 9 7 23.54 2 3.03 32.47 30.95 3.66 0 7 23.54 2 3.03 32.47 30.95 3.66 8 27.07 5 2 3.54 32.88 3.43 3.67 2 6 2 27.93 5 2 3.54 32.88 3.43 3.67 3 6 2 27.93 7 32.0 33.52 3.82 32.3 4 5 2 28.74 7 32.0 33.52 3.82 32.3 Table 7: Design summary optimization II. Baseline Design V Design VI Design VII Design VIII Σ RMS 3.7E-07 4.6E-08 2.E-08 2.E-07 3.7E-08 2004 WIT Press, www.witpress.com, ISBN -8532-77-5

524 High Performance Structures and Materials II 6 Concluding remarks Based on the modal frequency response (Optimization I), Design II is the best design obtained in using normal mode approach optimization, while Design VI is the best design obtained using the modal frequency response approach (Optimization II). However, in terms of mass increase relative to the baseline design, Design I and Design VII showed relatively lesser increase in their masses. In terms of lower increase in mass, and lower RMS summation of modal response Design V proved to be the best..e-03 Design V Design VI Design VII Design VIII Baseline.E-04.E-05 X displ (log scale).e-06.e-07.e-08.e-09.e-0 2.0 4.0 6.0 8.0 20.0 22.0 24.0 26.0 28.0 30.0 32.0 Frequency (Hz) Figure 7: Modal frequency response for baseline design. References [] Soedel, W., "Shell Vibration Without Mathematics Part I", S/V, Vol. 9, No., November 975, pp. 20-23. [2] Soedel, W., "Shell Vibration Without Mathematics Part II", S/V, Vol. 0, No. 4, April 976, pp. 2-7. [3] Alzahabi, B., Natarajan, L. K. Analytical Solution of Circular Cylindrical Shell Vibrations, Proceeding of the ISMA 2002 International Conference on Noise & Vibration Engineering, 6-8 September 2002, Leuven, Belgium. 2004 WIT Press, www.witpress.com, ISBN -8532-77-5

High Performance Structures and Materials II 525 [4] Alzahabi, B. Optimum Design of Submarine Hulls, Proceeding of the International Conference on High Performance Structures and Composites, Seville, Spain, - 3 March 2002, pp. 463-470. [5] MSC.visualNastran Quick Reference Guide, 200, MSC.Software, Santa Ana, CA, 200. [6] MSC/NASTRAN Optimization and Design Sensitivity, 200, MSC.Software, Santa Ana, CA, 200. [7] Alzahabi, B., Bernitsas, M.M., "Redesign of Cylindrical Shells by Large Admissible Perturbations", Journal of Ship Research, Vol. 45, No. 3, September 200, pp. 77-86. 2004 WIT Press, www.witpress.com, ISBN -8532-77-5