Scaling Law of a Damped Vibrating Rectangular Plate

Modern structural dynamics make use of a range of analysis and investigative approaches of different types such as: analytical, numerical, experimental or the combination of the three. The investigation in the laboratory, in particular, is crucial but unfortunately laboratory experiments present some limitations mainly imposed by the size. Dealing with large experimental structures can be time consuming and expensive. The concept of similitude is widely applied in the engineering vibroacoustic applications since it is particularly useful when applied to the testing of models with increasing complexity. The similitude schemes allow predicting the dynamic response of an original system by using the information obtained from a similar one, which is called avatar or replica according to the partial or complete degree of similitude, respectively defined as parent the original structure, a replica will be the physical system in complete similitude with the parent, an avatar will be the physical system in incomplete similitude with the parent. The prototype is called replica when the similitude is complete, all the geometrical parameters of the prototype are exactly scaled by the same factor. On the contrary, an avatar is the result of a distorted similitude when not all the geometrical parameters are scaled by a unique parameter. Using the Energy Distribution Approach for the study of the similitude is possible to demonstrate that when the damping between the model and the prototype is different, a replica cannot be obtained. There are different types of similitude: kinematic, dynamic, energetic and geometric similitude. The geometric similitude is particularly helpful because it allows avoiding measurements on original structures with big, or small, geometric sizes which can make laboratory experiments impractical or too costly. In addition to the geometric similitude, the focus on damping similitude has increased in recent years: this because the damping represents a highly relevant property during the design phase of any structure. There are different models of damping: the viscous damping is related to the dissipation of vibratory energy in solid mediums and structures over time and distance. It occurs whenever there is any type of friction that diminishes movement and disperses the energy and its corresponding force results to be proportional to the velocity. A contribution to the total damping of the system is also provided by the material itself, where energy dissipated in a vibrating material due to the intersections occurring at the molecular level: this contribution is described by loss factor η. This work presents an investigation on how to predict the frequency response of a system with a different structural damping (for this reason it will be an avatar) compared with the original one (parent). Such responses are determined retrieving a scaling law that includes this crucial dynamic characteristic η. This paper, after these remarks, in Section 2 a summary of the analytical study which will be considered as the starting point for defining the accuracy of the scaling law derived.