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en-USJuly 2012 en-US en-USwww.sname.org/sname/mt and frame method used in most steel and aluminum shipbuilding lends itself to mod -eling using plate and beam element types. Plate elements are used to model the plat -ing of the side shell, decks, bulkheads, and oors, as well as the webs of large stieners. Beam elements are used to model stan -chions, small stieners, combings, and the anges of large beams and girders. Using this modeling technique, the largest possi -ble elements (lowest possible mesh density) is defined by the stiffener spacing at any location within the vessel, as each stiff -ener is explicitly modeled. For most vessels, this leads to a mesh size of approximately 300-800 mm (12-32 in.). is size is important for two consider -ations: the applicability of plate elements and the usable frequency range. Plate ele -ments are not recommended when the thickness of the material to be modeled exceeds 1/10th of the element size, so plating thicknesses of up to 30 mm (1.2 in.) are gen -erally acceptable, with thicker plates being acceptable in some cases. e maximum frequency range of the model is determined by the element size as there needs to be a suf -cient number of elements in a wavelength of the structure at the frequency of interest. At a minimum, it is recommended that there be at least six elements per wavelength at the highest frequency. For plating, the waves of primary interest are bending (flexural) waves. The wave -length for these waves is dependent on the thickness and material of the plating, and may be a limiting factor for thinner plating, which will produce shorter wavelengths. Keep in mind, however, that often at very low frequencies the wavelength is signi -cantly aected by the stiness provided by the stiening elements, so classical equa -tions of bending wavelengths in plates alone would need to be modied to account for the composite stiness of the structure. Analysis type As mentioned earlier, there are two analy -sis types commonly used, natural frequency and forced response. Natural frequency analyses, also called normal modes or eigenvectors, is typically only used for the major hull modes when using a full ship model. Past the major hull modes, the modal density of a full ship model will become increasingly greater with frequency and many sections of the structure will have a large number of closely-spaced modes. For this reason a forced analysis is performed when frequencies of interest exceed 5-10 Hz. Forced analyses use forces applied at the various sources of vibration, primarily including forces generated by the propeller and propulsion machinery. is approach has the advantage of focusing the analysis only at the frequencies that will be excited by the ship?s machinery and will enable the modeler to make direct assessments of vibration levels throughout the mod -eled structure to determine if they are low enough to be acceptable. e primary goal of both analysis types is the avoidance of strong resonant modes with the excitation frequencies produced by equipment, primarily propulsion, on the vessel. For hull modes it is recommended that excitation frequencies should be at least ±10% of the frequency of lowest hull modes. As outlined earlier, at higher frequencies some sort of resonance coincidence is very likely, though the actual response of such modes may not be strong due to the loca -tion and magnitude of the forces applied to the structure at the source. is is a distinct advantage of the forced response approach, in that identication of signicant modes can be implicitly performed by assessing the magnitude of the structural response to any given excitation. Ultimately, the forced response analy -sis proved a method of assessing whether or not the vibration levels are within appro -priate levels. Of course, the forced response analysis requires additional informa -tion in order to get results, which include some knowledge (or estimate) of the forces applied to the structure (both in frequency and magnitude) as well as the damping of the structure. This information may not always be readily available. The results of both types of analyses can be seen in Figures 3 and 4. e natural frequency analysis produces mode shapes and frequencies that can be compared to excitation frequencies. e forced analysis produces vibration levels on the ship?s struc -ture that can be used to see local responses and for comparison to vibration criteria. To learn more about acoustic modeling, check out the following resources. International Maritime Organization Code on Noise Levels on board Ships - IMO 468(XII). Guide for Crew Habitability on Workboats, American Bureau of Shi pping (2008). United States Coast Guard, Recommendations on Control of Excessive Noise, Navigation and Vessel Inspection Circular No. 12-82 (1982). American Bureau of Shipping, Guide for Passenger Comfort on Ships (2001). International Standard Organization?s IS O-6954 ?Mechanical vibration ? Guidelines for the measurement, reporting, and evaluation of vibration with regard to habitability on passen -ger and merchant ships,? (2000). R. Lyon and R. DeJong, Theory and Application of Statistical Energy Analysis, Butterworth-Heinmann, (1995). FURTHER READING