THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

610 20. Vibration and Vibration Controlabout one-third the thickness of the structure or wall thickness. One thick layer onone side of the structure is generally more effective than two lesser layers on eitherside of the structure. This technique, which entails extensional damping, results inan economical, highly damped structure that is easy to fabricate.The mechanism of Figure 20.12(c) is essentially a sandwich arrangement, wherea viscoelastic layer is added to the structure. The covered (hence, constrained) viscoelasticlayer provides high extensional damping, but the entire structure becomesharder and more expensive to assemble.In order to be effective, the damping action must store a major portion of theenergy present in the entire system. Damping is best applied to points wherestretching or bending is the maximum, because these are the locations of maximalenergy storage.Viscoelastic layer techniques can be employed in a large variety of applicationsranging from walls, enclosures, barriers, conveyers, chutes, racks, and hoppers tothe most specialized, technologically sophisticated electronic instruments.20.7 Finite Element AnalysisWe have used relatively simple mathematical models to deal with vibration, butthe analysis of plates, shells, and other continuous systems can be difficult toanalyze without the aid of computers. Experimental modal analysis can providethe needed information, but this requires that a real structure be constructed andinstrumented to yield the desired data. Finite-element analysis (FEA), however,allows the problem to be represented with some detail and permits the designer tooptimize a design by investigating the effects of minor changes in the model uponthe static and dynamic states of the structure being evaluated. FEA is also used notonly to determine the statics and dynamics of beams, plates, shells, trusses, andother solid bodies but also to treat problems involving fluids, including airbornenoise propagation. A number of FEA programs can predict stresses and strains,temperature distribution stemming from heat and mass transfer as well as vibratorystates. Transient states are amenable to treatment by FEA. Since FEA is apowerful analytical tool, it no longer became necessary to fabricate a series ofactual structures before freezing a design. The most prominent programs containinggeneral codes for vibrational analysis include NASTRAN, developed by theU.S. National Aeronautics and Space Administration and now in public domain;MSC-NASTRAN r○ , a proprietary code developed from NASTRAN coding availablefrom MacNeal-Schwindler Corp.; and ANSYS r○ , another proprietary codethat is available from Swanson Analysis System, Inc.A structure undergoing analysis is modeled by subdividing it into various typesof finite elements from the finite-element library. The library may include more than100 element types, including beam elements, triangular and rectangular plate andshell elements, conical shell elements, and mass, damping, and stiffness elements.The preprocessing stage in FEA entails the creation of a finite-element mesh todepict the structure being evaluated. The elements may be automatically sized