Scaling laws obtained from a sensitivity analysis and applied to thin vibrating structures

Scaling laws are used to scale-up calculation or measurement results from a prototype structure to those of an original structure or vice versa. The scaling laws of mechanical structures are usually obtained from their equations of motion or from a dimensional analysis. This paper proposes scaling laws that are represented by a power law. The power law is directly deduced from the Π-theorem of similitude theory and the coefficients of the power law are obtained from first order sensitivities (sensitivity-based scaling laws). Among existing similitude analysis methods such as dimensional analysis the sensitivity-based scaling laws can be directly obtained from a model without a priori knowledge of its scaling behavior, e.g., from a finite element (FE) model. The applicability of the sensitivity-based scaling laws is demonstrated in three case studies. A simply supported rectangular plate subject to mechanical vibrations is modeled by an analytical approach based on the Kirchhoff plate theory and by an FE model based on the Mindlin-Reissner theory. The plate's dimensions are considered as design parameters for the sensitivity-based scaling laws. It is found in the first case study that the sensitivity-based scaling laws exactly predict the natural frequencies and the mean squared transfer admittances of scaled models from those of the original plate. This is in agreement with current literature and, thus, the method proposed in this paper can be considered verified. In the second case study the sensitivity-based scaling laws are directly obtained from the FE model. A good accuracy of the predicted vibration responses of the scaled models is found since (1) the scaled model and the original one are in similitude, i.e., the mode shape order and the damping are kept, and (2) the scaled plate can still be considered thin. The latter is assessed by the length (or width) to thickness ratios of the investigated plates and a lower bound is proposed. The third case study comprises a simplified car undercarriage to demonstrate the applicability of the method in practical design engineering. Besides the geometry parameters the material properties are incorporated in the sensitivity-based scaling laws as well. It is found that the natural frequencies as well as the vibration velocities at an arbitrarily chosen receiver point of the scaled models can be accurately predicted from those of the original one for complete similitude conditions.